red for the properties specially in dozenal (dependent in which base is used, in this wiki, we always use the dozenal base (i.e. base 10)).

## 0 to 100Edit

0 is the additive identity.

1 is the multiplicative identity.

2 is the only degree such that no magic squares exist.

3 is the smallest *k* such that it is impossible to construction of an angle equal to 1/*k* of a given arbitrary angle using only an unmarked straightedge and a compass.

4 is the smallest number of colors sufficient to color all planar maps.

5 is the smallest *k* such that general algebraic equation with degree *k* cannot be solved algebraically.

6 is the only degree (besides the trivial 2, there cannot be n mutually orthogonal Latin squares of order n for any n≥2) such that no Greek-Latin squares exist.

7 is the smallest *k* such that regular *k*-gon is not constructible using a compass and an unmarked straightedge.

8 is the smallest positive integer with no primitive roots.

9 is the only number *k* such that all even perfect numbers except 6 are of the form (*k* times a triangular number) plus 1 (i.e. centered *k*-gonal number). (besides, 9 is also the only number *k* such that (*k* times a triangular number) plus 1 (i.e. centered *k*-gonal number) is always also a triangular number. (thus, all even perfect numbers are triangular numbers) (thus, 9 is the only base such that all generalized repunits are also triangular numbers)

X is the smallest noncototient number.

E is the largest number which cannot be written as sum of two composite numbers.

10 appears in the value of the Riemann zeta function at −1 (i.e. ζ(−1) = −1/10, which equals the negative value of the reciprocal of 10, or the additive inverse of the multiplicative inverse of 10, this number is the value counter-intuitively ascribed to the series 1+2+3...).

11 is the largest number (and the only number beside 0 and 1) which cannot be written as sum of a positive squarefree number and a positive square.

12 is the smallest nontotient number.

13 is the smallest *k*>1 such that the number of terms of the *k*-th cyclotomic polynomial does not equal to the largest prime factor of *k*.

14 is the only number of the form *a*^{b} = *b*^{a} (commutative powers), with *a*, *b* nonnegative integers, *a* ≠ *b*.

15 is the largest number n such that there exist n real numbers 0 < a_1, a_2, a_3, ..., a_n < 1 such that all a_i*k are not integers for 1 ≤ i ≤ k and {floor(a_1*k), floor(a_2*k), floor(a_3*k), ..., floor(a_k*k)} = {0, 1, 2, ..., k−1} for all 1 ≤ k ≤ n,

16 is the largest number *n* such that the set of the integers 0≤*x*≤*n*−1 and gcd(*x*,*n*)=1, when mod any prime *p*, is not the complete set {0, 1, 2, ..., *p*−1}.

17 is the smallest number n such that Dirichlet character mod n containing numbers whose real and imaginary part are not constructible numbers.

18 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case.

19 is the smallest number of distinct squares needed to tile a square.

1X is the smallest nontrivial harshad number. (numbers that contain only one nonzero digit (i.e. numbers of the form *k*×10^{n} with 1≤*k*≤E and *n*≥0) are trivial harshad numbers)

1E is the smallest number *n* such that the relative class number h− of cyclotomic field Q(*e*^{2πi/n}) is greater than 1.

20 is the largest number such that 1 is the only quadratic residue coprime to it. (interestingly, if and only if a number *n* has this property, then *n* is divisor of 20).

21 is the smallest aspiring number (numbers not itself a perfect number whose Aliquot sequence terminates at a perfect number).

22 is the only positive number to be directly between a square and a cube.

23 is the number *n* for which (the largest number in the 3*x*+1 sequence starting at *n*)/(*n*^{2}) (i.e. 5414/(23^2) = 10.7E7314) is largest.

24 is conjectured to be the only number which can be expressed as a sum of the first positive integers (1 + 2 + 3 + 4 + 5 + 6 + 7), a sum of the first primes (2 + 3 + 5 + 7 + E) and a sum of the first nonprimes (1 + 4 + 6 + 8 + 9).

25 is the largest number *n* such that 2*x*^{2} + *n* is prime for all 0≤*x*≤*n*−1. (since it is divisible by *n* for *x* = *n*, one cannot do be better than this)

26 is the largest number with the property that all smaller numbers relatively prime to it are prime or 1.

27 is the smallest number *n* such that ʃ_{0}^{∞}(cos(*x*)cos(*x*/2)cos(*x*/3)...cos(*x*/*n*)) ≠ π/2.

28 is the smallest number *n* such that the *n*-th row of the modulo-2 Pascal's triangle (the top row, which contains only one 1, is the 0th row, not the 1st row), when read in binary, is not a number of the sides of a constructible regular polygon.

29 is the largest number that is not a sum of distinct triangular numbers.

2X is the smallest nonsquare number *n* (and the only such *n* ≤ 100, next such *n* is 102) not divisible by 4 with no prime factors *p* = 3 mod 4 but the period of continued fractions of √*n* is even.

2E is the smallest number *n*>1 such that gcd(*n*, *k*^{n}−*k*) = 1 is possible.

30 is the smallest perfect power which is not prime power.

31 is the smallest irregular prime.

32 is the magic constant of the only non-trivial normal magic hexagon.

33 is the smallest number *n*>2 such that the Mertens function *M*(*n*) returns zero.

34 is the largest number *n* such that almost all numbers reach *n* through the 3*x*+1 sequence. (about one half of numbers reach 11, and about one half of numbers reach 68, for the numbers ≤ 10000, E30X of them reach 34 (thus, E30E of them reach 18), 5788 of them reach 11, 5741 of them reach 68, 895 of them reach 28, and only 11 of them (the numbers of the form 3×2^{k}) reach 3)

35 is the largest number *n* such that *x*^{2} + *x* + *n* is prime for all 0≤*x*≤*n*−2. (since it is divisible by *n* for *x* = *n*−1, one cannot do be better than this)

36 is the largest integer a such that there exist integers b, c > 0 with 1/a + 1/b + 1/c = 1/2.

37 is the smallest number *n* such that (define *a*_{n}: *a*_{0} = 1, for *k* > 0, *a*_{k} = (1+*a*_{0}^{2}+*a*_{1}^{2}+...+*a*_{k−1}^{2})/*k*) *a*_{n} is not integer. (*a*_{n} is the Göbel's sequence) (the fractional part of *a*_{37} is 20/37)

38 is the smallest *n* such that all of *n*.0, *n*.1, *n*.2, ..., *n*.E (dot means concatenation) are composite. (i.e. all of 10*n*+0, 10*n*+1, 10*n*+2, ..., 10*n*+E are composite)

39 is the smallest odd number *n* that has more divisors than *n*+1 and that has a larger sum of divisors than *n*+1.

3X is the largest even number which is a value of *n* for incrementally largest values of minimal *x* satisfying the Pell equation *x*^{2}−*ny*^{2}=1.

3E (and the numbers of the form 3E×2^{k}, i.e. 7X, 138 and 274) is the only number *n* < 27E such that *n*×2^{k}+1 is composite for all 1≤*k*≤200 (in fact, all 1≤*k*≤400, the smallest *k*≥1 such that 3E×2^{k}+1 is prime is 407).

40 is the smallest number which is a member of betrothed number pair.

41 is the smallest number with the property that it and its neighbors are not squarefree.

42 is the smallest number which can be written as the sum of of 2 positive squares in 2 different ways.

43 is the smallest n such that both n−1 and n+1 are noncototients.

44 is the smallest untouchable number > 5 (5 is conjectured to be the only odd untouchable number).

45 is the smallest prime that produces prime reciprocal magic square.

46 is the smallest totient number which is not totient of squarefree number.

47 is the largest Fibonacci number which is also a triangular number.

48 is the only number *n* such that no *x*^{2} mod *n* is prime and *n* is not idoneal number. (the numbers *n* such that no *x*^{2} mod *n* is prime are 1, 2, 3, 4, 5, 8, 10, 13, 14, 20, 24, 34, 40, 48, 50, 60, 74, 94, X0, 120, 174, 180, 1E4, 220, 2X0, 374, 534, 5X0, 920, 10X0; and the idoneal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 13, 14, 16, 19, 1X, 20, 21, 24, 26, 29, 31, 34, 36, 39, 40, 49, 4X, 50, 5X, 60, 66, 71, 74, 79, 86, 89, 94, X0, XX, E1, 119, 120, 129, 13X, 156, 174, 180, 191, 1X9, 1E4, 220, 236, 249, 259, 281, 2X0, 326, 374, 534, 5X0, 920, 959, 10X0; 48 is the only number in the first sequence but not in the second sequence)

49 is the dimension of the smallest possible homogeneous space for *E*_{8}.

4X is the largest squarefree even number *n* such that the imaginary quadratic field Q(√−*n*) has class number 2.

4E is the smallest prime factor of the smallest composite Euclid number (i.e. 4E|(11#+1) = 15467 = 4E×365).

50 is the smallest possible order of nonsolvable group.

51 is conjectured to be the largest number *n* such that *kn*−1 and *kn*+1 are not both primes for all *k* ≤ 4*n*.

52 is the smallest number that can be written as the sum of of 3 distinct squares in 2 different ways.

53 is the largest number n such that no x^3 mod n is prime.

54 is the smallest number >1 which can be written as *k*^{p} with *p* prime in 2 different ways.

55 is the smallest deceptive prime.

56 is the denominator of the first Bernoulli number whose absolute value is not a unit fraction (*B*_{X} = 5/56).

57 is the smallest prime which is both Bernoulli irregular and Euler irregular.

58 is the smallest *n* which is not power of 10 and not congruent to 1 mod 11 (in which all such numbers are divisible by 11) such that *n*^{k}.1 (dot means concatenation) is composite for all 1≤*k*≤1000. (the smallest *k*≥1 such that this number is prime is 2781E5)

59 is the number in the square root of the algebraic form of the plastic number.

5X is the smallest weird number.

5E is the degree of Conway's polynomial.

60 is the smallest Achilles number.

61 is the largest squarefree number *n* such that the quadratic ring O_{Q(√n)} is a Euclidean domain.

62 is the number of different non-Hamiltonian polyhedra with a minimum number of vertices.

63 is the smallest n such that both n−1 and n+1 are nontotients.

64 is conjectured to be the smallest *n* such that there are no prime *n*-Fibonacci numbers. (the only smaller *n* with unknown status is 41)

65 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1.

66 is the dimension of the exceptional Lie group *E*_{6}.

67 is the smallest prime number *p* for which the real quadratic field Q[√*p*] has class number greater than 1.

68 is the number of single-tile moves required to optimally solve a Fifteen (13) game in the worst case.

69 is the only known square *n* such that *n*×2^{n}−1 is prime (Woodall prime). (note that its square root is also a square)

6X is the number of 6-hexes.

6E is the sum of the numbers that are not the sum of distinct triangular numbers.

70 is the smallest number *n* such that *n* is neither squarefree nor of the form *p*^{a}*q*^{b} with *p*, *q* primes, but no simple group with order *n* exists.

71 is the largest number *n* such that the sum of the first *n* positive square numbers is a triangular number.

72 is the smallest inconsummate number.

73 is the smallest imaginary quadratic field with class number 6.

74 is the smallest abundant number coprime to the smallest odd abundant number (669). (thus, 74 and 669 are the smallest coprime abundant numbers, since two even numbers cannot be coprime)

75 is the smallest prime to start a Cunningham chain of the first kind of ≥6 terms. (note that 2 starts a Cunningham chain of the first kind of 5 terms) (also note that there are no primes ≤ 400000 to start a Cunningham chain of the first kind of ≥7 terms, the smallest such prime is 46182E)

76 is the smallest number which is value of the Carmichael lambda function but not of the Euler totient function.

77 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed.

78 is the number of "atomic elements" in the Look-and-say sequence.

79 is the number in the square root of the algebraic form of the supergolden ratio.

7X is the smallest number *n*>1 such that *M*(*n*) is positive, where *M* is the Mertens function.

7E is the third-smallest number whose aliquot sequence terminates at 6 (within the sequence {7E, 21, 6}).

80 is the only number *n* beside 2 with the property that not *n* but *n*/2 is a value of *k* for incrementally largest values of groups of order *k* sets a record.

81 is conjectured to be the largest prime of the form 2^n+3^n.

82 is the only known number *n* of the form 2*p*^{2} with odd prime *p* such that Φ_{n}(2) is (probable) prime.

83 is the smallest number whose factorial is greater than googol (=10^{100}).

84 is the smallest Lagado number with more than one factorization into Lagado primes. (a Lagado prime is a Lagado number that is not divisible by any Lagado number between 1 and itself) (the Lagado numbers are the numbers = 1 mod 3 (i.e. end with 1, 4, 7 or X)

85 is the number of negated discriminants of orders of imaginary quadratic fields with 1 class per genus.

86 is the smallest number *n* for which (let *n* = 2^{a0} × 3^{a1} × 5^{a2} × 7^{a3} × E^{a4} × ...) a_{0} + a_{1}*x* + a_{2}*x*^{2} + a_{3}*x*^{3} + ... = 0 has solution, but does not have algebraic solution (i.e. there is no solution in radicals).

87 is the smallest Lucas-Wieferich prime associated with the pair (*P*, *Q*) = (4, 1). (the only other known such prime is 541670691)

88 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.

89 is the smallest integer such that the factorization of *x*^{n}−1 over *Q* includes coefficients other than ±1. In other words, the 89th cyclotomic polynomial, Φ_{89}, is the first with coefficients other than ±1.

8X is the denominator of an approximation of π (239/8X).

8E is the smallest prime *p* ends with E such that 2^{p}−1 is prime.

90 is the number of heptominoes (7-minoes).

91 is the number of different families of subsets of a three-element set whose union includes all three elements.

92 is the smallest number *n* such that there are no known powerful number *k* such that *k*+*n* is also powerful.

93 is the magic constant of the smallest 3×3 magic square using only 1 and prime numbers.

94 is the side of the smallest square that can be tiled with distinct integer-sided squares.

95 is the denominator of an approximation of π (257/95).

96 is the smallest integer not = ±4 mod 9 with no known 3-cube sum (there are no 3-cube sums for integers = ±4 mod 9, since cubes are = 0 or ±1 mod 9) (recently, the 3-cube sum of 29 and 36 were found). (the next such integer with no known 3-cube sum is 286, recently, the 3-cube sum of 119 was found)

97 is the smallest n such that the sum of the 5th powers of the proper divisors of n is prime.

98 is the final population for the Conway's game of Life starting with the "F-pentomino".

99 is the smallest possible length of the longest side of a Heronian tetrahedron (one whose sides are all rational numbers).

9X is the smallest *n* such that the range *n*, *n* + 1, ... 4*n*/3 contains at least one prime from each of these forms: 4*k* + 1, 4*k* - 1, 6*k* + 1 and 6*k* - 1.

9E is the largest number *n* such that the *n*th triangular number is also a tetrahedral number.

X0 is the smallest number to appear 6 times in Pascal's triangle.

X1 is the only Brazilian number with exactly 3 divisors.

X2 is the number of partitions of 20 into distinct parts.

X3 is the smallest *k* for which there is no known prime of the form (*k*−1)×*k*^{n}+1.

X4 is the smallest nontotient which is also an untouchable number.

X5 is the largest two-digit narcissistic number.

X6 is the number of different semigroups on 4 elements (up to isomorphism and reversal).

X7 is conjectured to be the largest number such that all three conditions of The New Mersenne Conjecture (*n* is of the form 2^n±1 or 4^n±3 (or both), 2^n−1 is prime, (2^n+1)/3 is prime) are true. (The New Mersenne Conjecture is that there is no number such that exactly two of these three conditions are true)

X8 is the largest number that is not a sum of distinct square numbers.

X9 is the smallest number that can be written as the sum of 3 squares in 4 ways.

XX is the only integer that is the sum of the squares of its first four divisors.

XE is the smallest Sophie Germain prime congruent to 3 mod 4 which is not safe prime.

E0 is the number of primary pretenders.

E1 is the smallest overpseudoprime base 10.

E2 is the smallest number whose aliquot sum is a weird number.

E3 is the product of the first two odd composite numbers.

E4 is the only number which is a self-descriptive number in some base (base 4) which has a smaller self-descriptive number (84).

E5 is the smallest strictly non-palindromic number *n*>4 such that *n*+2 is also strictly non-palindromic.

E6 is the smallest number whose aliquot sequence has length >20 (in fact, >120, its length is 12X).

E7 is the largest prime factor among the smallest pair of odd amicable numbers (7139 and 8543).

E8 is the largest number whose square is a tetrahedral number.

E9 is the smallest *n*>1 such that *n*×2^{n}+1 is prime (Cullen prime).

EX is the number of planar graphs with 6 unlabeled vertices.

EE is the only number which is product of twin primes but not brilliant number.

100 is the largest Fibonacci number which is also a square number.

## 101 to 1000 (selected)Edit

101 is the smallest fundamental discriminant of real quadratic number fields with class number 4.

102 is the smallest untouchable number which is semiprime. (also the smallest untouchable number >2 which is = 2 mod 4)

103 is the number of sided 6-hexes.

104 is the largest number *n* such that the primitive part of 2^{n}+1 was once the largest known prime. (start with 2^{375}−1, the largest known prime is almost always Mersenne prime, i.e. of the form 2^{n}−1 (not +1), the only one exception is 16X739×2^{X5141}−1)

105 is the smallest number which is not sum of two prime powers (including 1).

106 is the smallest unitary admirable number which is not squarefree.

107 is the smallest emirp containing at least one zero digit.

108 was once the smallest base not of the form n^x (where generalized repunits can be factored algebraically) for which no generalized repunit (probable) primes are known (currently, the smallest such base is 135). (recently, the probable prime (108^110461−1)/107 was found)

109 is the sum of the first 5 positive factorials.

10X is the number of regions in regular nonagon (9-gon) with all diagonals drawn.

10E is the sum of the primes between its smallest and largest prime factor.

110 is the smallest number that is the product of two different substrings.

111 is the smallest irregular prime with irregular index greater than 1.

112 is the largest base *b* for which there are no primary pretenders <*b*.

114 is the smallest number *n* with exactly 10 solutions to the equation φ(*x*) = *n*.

115 is the smallest Harshad number >10 divisible by neither E nor 10.

116 is the triple factorial of 9 (3×6×9).

117 is the largest Heegner Number.

119 is the midpoint of the *n*th larger prime and *n*th smaller prime for all 1≤*n*≤6.

11X is conjectured to be the largest even panconsummate number.

11E is the only prime requiring exactly 8 cubes to express it.

120 is the smallest possible order of noncyclic simple group other than groups of the form *A*_{k} (which is always a noncyclic simple group for *k*≥5).

121 is the largest Pell number which is also a square number.

122 is the smallest even number *n* such that the smallest *N* such that the multiplicative group of integers modulo *N* ([*Z*/*NZ*]^{×}) as a product of cyclic groups *C*_{k1} × *C*_{k2} × ... × *C*_{km} contains a copy of *C*_{n} has *m* > 2. (the corresponding *N* is 624451, and the multiplicative group of integers modulo 624451 ([*Z*/624451*Z*]^{×}) is isomorphic to *C*_{2} × *C*_{122} × *C*_{2440})

123 is the only Smarandache number which is also triangular number.

125 is the smallest 3-digit Keith number.

127 is the smallest n>1 such that n^6+6 is prime.

128 is the smallest number >1 which is both pentagonal (5-gonal) and octagonal (8-gonal).

129 is the magic constant of the smallest 3×3 magic square using only prime numbers.

12X is the smallest number which is nontrivially palindromic in three or more consecutive integer bases (base 6, base 7, and base 8). (single-digit numbers are trivially palindromic)

12E is the smallest nonpalindromic number whose square is palindromic.

130 is the largest possible number of edges of an Archimedean solid.

131 is the largest value *x* satisfying the Ramanujan–Nagell equation (2^{n}−7 = *x*^{2}).

133 is the largest number that equal the sum of the squares of the digits of their own square.

135 is the smallest composite primeval number.

138 is the number of semigroups with order 4.

13E is the smallest prime congruent to 1 mod 17 (this is conjectured to be the case which (let *a*(*n*) is the smallest *k* such that *kn*+1 is prime) log_{n}(*a*(*n*)−1) is largest (i.e. log_{17}(9) = 0.8E55967E072E...), it is also conjectured that log_{n}(*a*(*n*)−1) is always < 0.9, note that log_{n}(*a*(*n*)−1) < 0.8E does not work, and log_{17}(9) = 0.8E55967E072E... is exactly the counterexample, it also has been conjectured that log_{n}(*a*(*n*)−1) < 1). (note that 17 is the smallest primitive root mod 13E, this is the second-largest case which smallest primitive root mod *p* (with *p* prime) is larger than √*p*, the largest case is the smallest primitive root mod 2X1 is 19)

140 is the smallest multiple of 10 which is not Harshad number.

141 is the largest number that can be written as *ab* + *ac* + *bc* with 0 < *a* < *b* < *c* in a unique way.

142 is the number of irreducible representations of the Monster group.

143 is the smallest *n* such that binomial(2*n*, *n*) is divisible by *n*^{2}.

145 is the smallest prime *p* such that none of 2*p*+1, 4*p*+1, 8*p*+1, X*p*+1, 12*p*+1, and 14*p*+1 is prime. (Sophie Germain proved that Fermat's last theorem is true for all odd primes *p* such that at least one of 2*p*+1, 4*p*+1, 8*p*+1, X*p*+1, 12*p*+1, and 14*p*+1 is prime)

147 is the largest *k* such that all positive values of *k*−2*n*^{2} are primes or 1.

148 is the largest number *n* ≤ 100000000 such that |*M*(*n*)| ≥ (√*n*)/2, where *M* is the Mertens function. (Mertens conjectured that |*M*(*n*)| < √*n* for all *n* > 1, this is now known to be false)

149 is the smallest number whose square is de Polignac number.

14X is the number of distinct (non-isomorphic) directed graphs on four unlabeled vertices, not having any isolated vertices.

14E is the number of integer squares (not necessarily of unit size) can be found in a staircase-shaped polyomino formed by stacks of unit squares of heights ranging from 1 to 10.

150 is the smallest non-semiprime whose square is a triangular number.

151 is the smallest nonsquare number *n* not divisible by 4 and with no prime factors *p* = 3 mod 4 but the period of continued fractions of √*n* is even.

155 is the smallest Chebyshev pseudoprime base 2.

156 is the largest number *n* such that all primes between *n*/2 and *n* yield a representation as a sum of two primes.

157 is the smallest primorial prime which is not from twin primes.

160 is conjectured to be the only number not of the form *t*+*p*, with *t* triangular number (including 0 and 1), *p* either prime or 0.

161 is conjectured to be the largest number *n* such that σ(*n*)−*n* is odd but there are no *k* ≠ *n* such that σ(*k*)−*k* = σ(*n*)−*n*.

162 is conjectured to be the largest composite number *n* such that omega(10^{n}−1) = d(*n*) (where omega(*n*) is the number of distinct prime factors of *n*, d(*n*) = σ_{0}(*n*) is the number of divisors of *n*)

163 is the number of space groups, not including handedness.

164 is the smallest number which is a member of amicable number pair.

165 is the smallest odd nonsquare number *n* such that *x*^{2}−*ny*^{2} is not solvable, but *x*^{2}−*2ny*^{2} is.

167 is the only number that cannot be written as a sum of 30 or fewer fifth powers.

169 is the smallest number *n* besides 1 and 9 for which σ(φ(*n*)) = φ(σ(*n*)).

16E is a quadratic nonresidue mod every number 3≤*n*≤24, besides, 16E is also a primitive root mod every number 1≤*n*≤16 which have a primitive root.

170 is the smallest even number *n* such that the numerator of the *n*th Bernoulli number is divisible by a nontrivial square number that is relatively prime to *n*.

171 is the smallest fundamental discriminant of real quadratic number fields with class number 3.

172 is the number of space groups, including handedness.

173 is the smallest number with ≥3 odd prime factors whose cyclotomic polynomial has all coefficients ±1.

174 is the number of digits of 100!.

175 has a palindromic reciprocal: 0.0074EE470074EE4700...

179 is conjectured to be the smallest Lychael number.

17X is the length of the longer of the two ladders such that all of the distance of two buildings, the lengths of the two ladders, and the height of the cross point of two ladders are integers, and the height of the cross point of two ladders is minimal. (in this case, the length of the shorter of the two ladders is 95, the distance of two buildings is 94, and the height of the cross point of two ladders is 12)

17E appears in one of the earliest known geometrically converging formulas for computing pi: pi/4 = 4*arctan(1/5) − arctan(1/17E).

180 is the kissing number in 8 dimensions. (this is the *E*_{8} lattice) (note that the true value of the kissing number is only known in 1, 2, 3, 4, 8, and 20 dimensions)

181 is the smallest (and the only known) 3-Wall-Sun-Sun prime.

182 is the smallest *n* such that *n*, *n*+1, *n*+2, and *n*+3 have the same number of divisors.

183 is the smallest Frugal number.

184 is the smallest number divisible by 4 which is neither totient nor cototient.

186 is the smallest number n for which it is known that there is an infinite number of prime gaps no larger than n.

187 is the smallest n such that both n−1 and n+1 are untouchable.

188 is the smallest number n>1 for which the arithmetic, geometric, and harmonic means of φ(n) and σ(n) are all integers.

18E is the smallest number that can be formed in more than one way by summing three positive cubes.

190 is the 5th central binomial coefficient.

191 is the smallest non-trivial triangular star number.

193 is the smallest cyclic number which is neither prime nor semiprime.

194 is the value of ^{2}4 (where ^{n}*m* is the tetration).

195 is the only known Fermat prime which is irregular prime.

196 is the magic constant of the smallest 4×4 magic square using consecutive prime numbers.

198 the magic constant of a normal 8×8 magic square.

199 is the number of octacubes (8-cubes) that are unfoldings of the tesseract.

19E is the smallest prime *p* such that (*p*−1)/2 is irregular prime. (note that 19E itself is also irregular prime)

1X0 is the largest number *n* such that carmichael_lambda(*n*) = X.

1X1 is the value of !6 (= 6!(1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5! + 1/6!)).

1X3 is the number of groups with order 2^{6} (=54).

1X5 is the smallest prime base for which no generalized repunit (probable) primes are known.

1X7 is the smallest prime *p* such that neither *p*−1 nor *p*+1 is cubefree.

1X8 is conjectured to be the largest negative number appearing in a cycle of Collatz (3x+1) conjecture.

1XE is the smallest non-semiprime n such that gcd(n, 2^n - 2) = 1. (also the smallest non-semiprime n such that n^3 divides Sum_{k=1..n-1} k^n)

1E0 is the smallest number whose aliquot sequence has not yet been fully determined.

1E1 is the 8th Euler (or up/down) number.

1E3 is the smallest odd number n such that no (probable) primes of the form ((n+1)/2)^k − ((n−1)/2)^k are known.

1E4 is the base with the largest conjectured smallest generalized Sierpinski number and the largest conjectured smallest generalized Riesel number in all bases ≤1000 (this two numbers are in order 924XE44391X56 and 49731795912E69) (only the numbers with covering sets of >1 primes are considered as generalized Sierpinski/Riesel numbers, the numbers with a trivial factor and the numbers that make a full covering set with all or partial algebraic factors are not considered as generalized Sierpinski/Riesel numbers). (the next base with larger conjectured smallest generalized Sierpinski number and larger conjectured smallest generalized Riesel number is 1254)

1E5 is the largest prime p such that (1!+2!+3!+4!+ ... +p!) − 2 is prime.

1E7 is the smallest *n* such that φ^{7}(*n*) > 1.

1E8 is the smallest *n* appearing twice in *P* union *Q* union *R* defined with: Construct sequences *P*, *Q*, *R* by the rules: *Q* = first differences of *P*, *R* = second differences of *P*, *P* starts with 1, 3, 9, *Q* starts with 2, 6, *R* starts with 4; at each stage the smallest number not yet present in *P*, *Q*, *R* is appended to *R*.

1EX is the smallest nonsemiprime which is a possible value of the smallest (Fermat) prime base *n*.

200 is the smallest *n*>8 such that both *n* and *n*+1 are powerful.

201 is conjectured to be the largest square number which is one more than a number which is two times a power of 10 (2×10^{2}). (21 and 201 are both square numbers, it is conjectured there are no other square numbers of the form 2000...0001)

202 is the smallest *n* such that a positive definite integral quadratic form is universal if it takes the numbers from 1 to *n* as values. (a more precise version states that, if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, X, 11, 12, 13, 15, 17, 19, 1X, 1E, 22, 25, 26, 27, 2X, 2E, 31, 36, 4X, 79, 92, 101, 14E, 202, then it represents all positive integers)

203 is the largest number that is not the sum of distinct non-trivial powers.

204 is the 5th term of the continued fraction of π.

208 is conjectured to be the smallest *n* which is not power of 10 and not congruent to 1 mod 11 (in which all such numbers are divisible by 11) and not congruent to EX mod EE (in which all such numbers are divisible by either E or 11) such that (*n*^{k}.1) (dot means concatenation) is composite for all *k*≥1 (the only smaller *n* with unknown status are 117 and 153)

210 is the smallest nonsquare number which is not a primitive root mod any safe prime.

211 is the smallest 6-hyperperfect number.

214 is the 6th primitive abundant number.

215 is the smallest odd number *n*≥3 such that there are no known prime of the form *n*^{k}−2 with *k*≥1.

216 is the near-integer of pi^5.

217 is the smallest prime = 1 mod 3 for which 2 and 3 are cubic residues.

21E is the smallest *n* such that the repunit with length *n* has no known proper factors (except 1).

221 is the number of intersections when all the diagonals of a regular dozagon are drawn.

222 is the smallest happy number which is not power of 10.

223 is the smallest odd number *n* such that φ(*n*) < φ(*n*−1).

224 is the sum of all Heegner numbers.

225 is the smallest Luhn prime.

226 is the smallest number that start an unbounded aliquot-like sequence based on Dedekind psi function.

228 is the smallest *n*>0 such that 35*n*^{2}+1 is square.

22E is the smallest Fibonacci U-pseudoprime.

230 is conjectured to be the smallest negative base which is not perfect power such that there are no repunit primes. (the smaller bases with unknown status are 81, 87, 95, 136, 137, 164, 214, 216, and 219)

231 is the only known 3-hyperperfect number.

23E is the number of degree 10 irreducible polynomials over GF(2).

241 is the smallest odd number *n* such that |2^{k}−*n*| is composite for all 1≤*k*≤*n*.

245 is the smallest (Fermat) pseudoprime base 2 (also called Sarrus number or Poulet number).

247 is the only cube which is a nontrivial repunit in some base (base 16).

253 is the smallest n such that n±1 and n±3 are all abundant.

255 is the smallest number whose 4th power can be written as the sum of four 4th powers.

256 is the sum of the absolute values of the coefficients of Conway's polynomial.

260 is the smallest order of simple group which is a multiple of a smaller order of simple group (50).

262 is the smallest even base for which no generalized Carol primes are known.

265 is the smallest number that can be written as a sum of consecutive squares in more than 1 way.

269 is the number of octominoes (8-minoes).

271 is conjectured to be the largest prime *p* such that (previous prime(*p*))# ± *p* are both primes.

274 is the smallest generalized Riesel number base 10.

275 × 5*d* is 4-digit repdigit *dddd* for nonzero digit *d*.

276 is the smallest *n* ≠ X mod E and *n* is not power of 10 for which there are no non-titanic prime of the form *n*×10^{k}+1 with *k* ≥ 1.

277 is the largest known prime *p* such that *p*! − 1 and *p*# + 1 are both primes.

278 is the smallest even base not of the form n^x (where generalized repunits can be factored algebraically) for which no generalized repunit (probable) primes are known.

27E is the largest base *b* for which there are no primary pretenders <*b*−1.

280 is the order of the hyperoctahedral group for *n* = 4.

281 is the smallest integer such that the factorization of *x*^{n}−1 over *Q* includes coefficients other than ±1 and ±2.

290^{2n} appears in a denominator of an infinite product of π.

293 is the smallest Lucas-Carmichael number.

294 is conjectured to be the largest perfect power which is repunit with three or more digits in some base (base 7). (the only other two known such numbers are X1 and 247)

295 is the smallest fundamental discriminant of real quadratic number fields with class number 5.

298 is the *n* for which the smallest prime of the form *n*×10^{k}+1 with *k* ≥ 1 is largest for all *n* < 375 (the smallest generalized Sierpinski number base 10).

2X1 is conjectured to be the largest prime *p* whose smallest primitive root is larger than √*p*.

2XX is the smallest non-primepower *k* such that binomial(2*k*, *k*) = 2 (mod *k*). (besides, 2XX is also the only known such even non-primepower *k*)

2E0 is the largest number n which is divisible by all numbers less than or equal to the cube root of n. (the numbers n which is divisible by all numbers less than or equal to the square root of n are exactly the divisors of 20)

2E1 is the largest number = 1 mod 4 which is a value of *n* for incrementally largest values of minimal *x* satisfying the Pell equation *x*^{2}−*ny*^{2}=±1 (both +1 and −1 are allowed).

2E7 is the largest imaginary quadratic field with class number 2.

2EE is the smallest prime *p*>E ends with E such that the period length of 1/*p* is not (*p*−1)/2.

300 is the only number k besides −1 (which corresponds to the Catalan conjecture) such that the Mordell curves x^2+k=y^3 (which are specific elliptic curves) has rational solutions with both x and y nonzero, but only finitely many rational solutions (only consider primitive k, i.e. sixth-power-free k).

309 is the smallest Hilbert number with more than one factorization into Hilbert primes. (a Hilbert prime is a Hilbert number that is not divisible by any Hilbert number between 1 and itself) (the Hilbert numbers are the numbers = 1 mod 4 (i.e. end with 1, 5 or 9)

315 is the number of superabundant numbers which are also highly composite numbers.

319 is the number requiring the largest positive base (342863E) to be the primary pretender.

31X is the largest number that cannot be written as a sum of 7 or fewer cubes.

31E is the smallest n with omega(n)>2 such that gcd(n, 2^n - 2) = 1. (also the smallest n with omega(n)>2 such that n^3 divides Sum_{k=1..n-1} k^n)

321 is the smallest prime which is concatenation of numbers from some number (3) down to 1.

324 is the largest n such that (1!+2!+3!+4!+ ... +n!) − 2 is prime.

326 is conjectured to be the largest base *b* for which there are no (Fermat) pseudoprimes ≤*b*+1.

32E is the smallest prime *p* which divides (1!+2!+3!+4!+ ... +p!) − 2.

330 is the largest module for the known property of odd perfect numbers. (the known property of odd perfect numbers is = 1 mod 10, or = 99 mod 330, or = 69 mod 230)

335 is the smallest composite k such that 1^(k-1) + 2^(k-1) + 3^(k-1) == n (mod k).

340 is the largest number *n* such that carmichael_lambda(*n*) = 8.

344 is the smallest square number which is nontotient.

346 is the smallest nice Friedman number.

34E is the smallest irregular prime with irregular index greater than 2.

350 is the smallest number of faces such that holyhedron is known to exist.

353 is the largest sum-product number.

354 is the third perfect number.

355 is the smallest number k such that (2^k-2)*(k+1)+k is prime.

358 is the smallest strong Achilles number.

360 is the largest number *n* such that carmichael_lambda(*n*) = 6.

368 is the only known cube *n* such that *n*×2^{n}−1 is prime (Woodall prime). (note that its cube root is also a cube)

369 is the smallest nonsquare automorphic number.

373 is the number with the lowest property (for random base) to be the primary pretender.

375 is the square root of the smallest Perrin pseudoprime. (note that the smallest Perrin pseudoprime (the square of 375) is 111101, a near-repunit number, and contains only five 1's and one 0, no any digit >1)

378 is the starting number of the first run of 15 (or 12) consecutive composite numbers.

380 is the smallest number which cannot be made prime by changing one of its digits.

383 is the smallest *n*>0 such that 37*n*^{2}+1 is square.

385 is the smallest n such that both n−1 and n+1 are both nontotients and noncototients.

388 is the smallest number > *e*^{2π}.

38X is the number requiring the largest negative base (2516X5X) to be the primary pretender.

390 is the only n with ≥2 digits that are equal to the sum of descending numbers raised to their digits powers (i.e. 390 = 3^3 + 2^9 + 1^0). (Note that there are no n with ≥2 digits that are equal to the sum of ascending numbers raised to their digits powers (i.e. n = *abcd...* = 1^*a* + 2^*b* + 3^*c* + 4^*d* + ...), neither n with ≥2 digits that are equal to the sum of their digits raised to descending number powers (i.e. n = *...dcba* = ... + *d*^4 + *c*^3 + *b*^2 + *a*^1), but there are two n with ≥2 digits that are equal to the sum of their digits raised to ascending number powers (XE = X^1 + E^2, and 55981002E = 5^1 + 5^2 + 9^3 + 8^4 + 1^5 + 0^6 + 0^7 + 2^8 + E^9))

3X7 is the largest number that cannot be written as a sum of 16 or fewer fourth powers. (all numbers can be written as a sum of 17 or fewer fourth powers)

3X8 is the smallest number which is a Rhonda number in some base (base 10).

3X9 is the smallest Carmichael number.

3XX is the smallest number not itself an amicable pair whose Aliquot sequence terminates at an amicable pair.

3XE is the largest known Wilson prime. (the only other known such primes are 5 and 11)

3E8 is conjectured to be the largest base *b* for which there are no (Fermat) pseudoprimes ≤*b*−1.

3E9 is the smallest (Fermat) pseudoprime base 3E8 (the largest base *b* for which there are no (Fermat) pseudoprimes ≤*b*−1).

404 is the smallest number *n* such that φ(*x*) = *n* has only two solutions and the smaller of this two solutions is not prime power (including 1).

405 is the smallest number *n*≥2 such that there are no known prime of the form 2×*n*^{k}−1 with *k*≥1.

407 is the smallest n such that 3E×2^{n}+1 is prime.

41X is the smallest even Quasi-Carmichael number.

420 is the largest possible number of cells of 4-dimention polytope. (note that for *n*-dimention polytope, *n*≥5, the only possible number of cells are *n*+1, 2*n*, and 2^{n})

42E is the smallest number whose square is an even-digit palindromic number.

43X is the largest number *n* such that *n*, *n*^{2} and *n*^{3} are all distinct-digit numbers.

442 is conjectured to be the largest number *n* such that there is no 0≤*k*≤sqrt(*n*)*ln(*n*) such that both *n*+*k* and *n*+*k*^{2} are primes. (the other such *n* are 1, 19, 232, 245)

446 is the smallest unitary admirable number which is not primitive unitary abundant number (unitary abundant numbers having no unitary abundant proper unitary divisor).

455 is the smallest prime which is a prime factor of a composite Fermat number.

459 is the smallest nonsemiprime which is not Carmichael number and is a possible value of the Euler primary pretender to base *n*. (note that the only nonsemiprime which is a possible value of the primary pretender to base *n* is the smallest Carmichael number (3X9))

460 is the smallest base such that the smallest (Fermat) pseudoprime is nonsemiprime (281).

46E is the smallest prime factor of the smallest composite repunit with prime length.

470 is the smallest *n* (and the only *n*≤1000) such that *k*×*n* is Harshad number for all *k*≤600000. (the smallest *k* such that 470*k* is not Harshad number is 750275)

497 is the first irregular prime to appear in the numerator of a Bernoulli number.

4X1 is the smallest number n such that n^3 divides Sum_{k=1..n-1} k^n but gcd(n, 2^n - 2) != 1. (note that there are no numbers n such that gcd(n, 2^n - 2) = 1 but n^3 does not divide Sum_{k=1..n-1} k^n)

4X4 is the smallest base such that the smallest (Fermat) pseudoprime is Carmichael number (3X9).

4X5 is the length of the largest proved repunit prime.

4X7 is the smallest n which is strong pseudoprime to base b, where b is the smallest number such that Jacobi(b|n) = −1.

4X9 is the smallest Fibonacci V-pseudoprime.

4E6 is conjectured to be the largest number *n* such that *n*×(*n*+1) is a primorial (15#).

4E7 is the smallest number k such that (2^k-2)*(k+1)-k is prime.

500 is the smallest number which cannot be written as *mn* with both *m* and *n* are powers of squarefree numbers (including 1).

520 is the constant term of modular function j as power series in q=e^(2\pi i t).

53E is the smallest number m where there is not a k such that C^{k}(m) = 1, where C(m) = m/10 if m == 0 (mod 10) and C(m) = 10*(m + floor(m/10) + 1) otherwise. C^{0}(m) = m and C^{k}(m) = C(C^{k - 1}(m)).

545 is the smallest odd number *n* such that 2^{k}+*n* is composite for all 1≤*k*≤*n*.

551 is the period of the sequence of Bell numbers mod 5.

555 is conjectured to be the largest repdigit number which is one more than a square number (554=24^{2}).

564 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 43 with 0 <= |x| <= |y| <= |z|.

573 is conjectured to be the smallest number *n* divisible by 3 such that *n*×2^{k}−1 and *n*×2^{k+1}−1 are not both primes for all *k*≥1. (currently, the smallest such *n* with unknown status is 33)

598 is the smallest weird number which is also an untouchable number.

5X0 is the largest number *n* such that *k*^2 mod *n* is square number for all *k* coprime to *n*.

5E6 is the Kaprekar constant for 3-digit numbers.

611 is the largest known (p, p−9) irregular prime. (the only other known such prime is 57)

614 is the number of 4×4 magic squares.

620 is the starting number of the first run of 17 (or 16) consecutive composite numbers.

624 is the smallest number *k* such that *nk*+*x*, where *x* runs through the set of the integers 0≤*x*≤*n*−1 and gcd(*x*,*n*)=1, are all primes, for *n* = 16, the largest number *n* such that the set of the integers 0≤*x*≤*n*−1 and gcd(*x*,*n*)=1, when mod any prime *p*, is not the complete set {0, 1, 2, ..., *p*−1}. (such number *k* is 2 for *n* = 1 and 1 for all other such number *n*: 2, 4, 6, X and 10)

635 is the smallest composite de Polignac number.

637 is the largest imaginary quadratic field with class number 3.

63E is conjectured to be the largest number n such that there is no k≤n such that k*n is practical and k*n±1 are twin primes.

641 is the only composite n ≤ 10^{8} such that V_{n+1} = 2Q (mod n), where D, P, and Q are chosen by Selfridge's method.

650 is the 6th central binomial coefficient.

666 is the "beast number".

668 is the smallest 3-digit narcissistic number.

669 is the smallest odd abundant number.

66X is conjectured to be the largest even number which is (Fermat) pseudoprime to 1/4 of the bases coprime to it.

673 is the smallest (Fermat) pseudoprime base 326 (the largest base *b* for which there are no (Fermat) pseudoprimes ≤*b*+1).

67E is the smallest number k such that (2^k-2)*(k-1)-k is prime.

693 is the sum of proper divisors of the smallest odd abundant number (669).

695 is the smallest non-Fermat prime p such that phi(p-2) = phi(p-1).

6X5 is the smallest extra strong Lucas pseudoprime.

6X7 is the largest number which cannot be expressed as the sum of abundant numbers.

6X9 is conjectured to yields the highest residue for 2^h(n)/(3^t(n)*n), where h and t are the number of halving resp. tripling steps in the Collatz problem.

6E7 is the largest number that cannot be written as a sum of admirable numbers.

703 is the smallest (Fermat) pseudoprime base 704 (the largest base *b* for which there are no (Fermat) pseudoprimes <*b*−1).

704 is conjectured to be the largest base *b* for which there are no (Fermat) pseudoprimes <*b*−1.

70E is conjectured to be the largest odd number not of the form p^2 + q^2 + r with p, q, and r primes.

721 is the smallest possible sum of two coprime abundant numbers.

75E is the smallest integer > 1 congruent to +1 or −1 modulo k for all 1 ≤ k ≤ 10. (note that 75E is not prime (it is divisible by 10+1 (=11)), but the smallest integer > 1 congruent to +1 or −1 modulo k for all 1 ≤ k ≤ n is prime for all 1 ≤ n < 10)

771 is the smallest Wieferich prime.

777 is the smallest number with multiplicative persistence 5.

77E is the number of steps for the Conway's game of Life starting with the "F-pentomino" to stabilize.

780 appears in the aliquot sequence of 780/4 (=1E0), which is the smallest number whose aliquot sequence has not yet been fully determined.

781 is the smallest deceptive prime which is not semiprime.

782 is the smallest number *n* which is (Fermat) pseudoprime to exactly 11 bases 0≤*b*≤*n*−1. (note that for all 1≤*k*≤11, but not for *k*=12, there exists number *n* which is (Fermat) pseudoprime to exactly *k* bases 0≤*b*≤*n*−1, and for *k*=11, the smallest such number *n* is largest for all these values of *k*)

7X2 is the starting number of the first run of 19 (or 18) consecutive composite numbers.

853 is the smallest n such that both n and n+1 are exceptional numbers.

861 is the smallest number >1 which is both square and hexagonal (6-gonal).

87E is the smallest *N* such that the multiplicative group of integers modulo *N* ([*Z*/*NZ*]^{×}) as a product of cyclic groups *C*_{k1} × *C*_{k2} × ... × *C*_{kn} contains a copy of *C*_{n}, where *n* is the smallest even number not in the range of Carmichael lambda function (*n*=12).

890 is the smallest number ≥4 not of the form 2^m + n! + p with m≥0, n≥0, p prime.

8X4 is the smallest *n*>7 such that *n*! is not Harshad number.

8X7 is the largest exponent ≤1000 of Mersenne prime.

916 is the sum of the numbers that are not the sum of distinct square numbers.

928 is the starting number of the first run of 29 consecutive composite numbers. (no other such number <4X98)

X00 is the smallest n>14 (also the smallest n which is not power of 2) such that there exist k such that 1^n+2^n+3^n+...+k^n is prime (k=5).

X45 is conjectured to be the largest Stern prime.

X65 is the smallest Quasi-Carmichael number to >2 bases. (the smallest Quasi-Carmichael number is 2E, and the smallest Quasi-Carmichael number to >1 bases is 165)

X83 is the only 3-digit narcissistic number with distinct digits.

X85 is the smallest *n* such that the number of bases *b* mod *n* for which *b*^{n−1} = 1 mod *n* is a nontotient (344).

X91 is the smallest number >20 which is not the sum of a perfect power (not including 0 and 1) and a prime.

X97 is the largest imaginary quadratic field with class number 4.

XXE is the smallest number with multiplicative persistence 6.

XE3 is the smallest odd abundant number which is not admirable number.

E11 is the only number k < 41013 (the smallest k such that gcd(k−1, 6−1) = 1 and k×6^{n}−1 is composite for all n≥1) such that gcd(k−1, 6−1) = 1 and no primes of the form k×6^{n}−1 with n≥1 are known.

E15 is the constant term of *n*^{2} − 67*n* + E15, the only known quadratic polynomial which generates primes for all 0≤*n*≤67. (there are no known quadratic polynomials which generates primes for all 0≤*n*≤68, and conjectured not to exist)

E36 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 14 with 0 <= |x| <= |y| <= |z|.

E60 is the smallest even base for which no generalized Carol primes or generalized Kynea primes are known.

E71 is the smallest prime which is at the end of an arithmetic progression of 8 primes (147, 2X1, 437, 591, 727, 881, X17, E71).

E80 is the largest number *n* such that *k*^2 mod *n* is prime power for all *k* coprime to *n*.

E87 is the smallest Euler-Jacobi pseudoprime (also the smallest strong pseudoprime) base 13.

EE0 is the largest number in the 11st row of Pascal's triangle.

EE3 is the smallest de Polignac number divisible by 3.

EE6 is conjectured to be the largest number *n* which is not quadratic residue mod all primes *p*≤√*n* not dividing *n*.

1000 is the number of directed Hamiltonian paths in the 5×5 knight graph. (note that there are no directed Hamiltonian paths in the n×n knight graph for n<5)

## The famous Hardy-Ramanujan number 1001Edit

1001 is the smallest number which can be written as the sum of of 2 positive cubes in 2 different ways.

1001 is the smallest 4-digit palindromic number.

1001 is the smallest absolute Euler pseudoprime.

1001 is the smallest palindromic number which cannot be prime when read in any base.

1001 is the smallest Carmichael number of the form (6*n*+1)×(10*n*+1)×(16*n*+1) with 6*n*+1, 10*n*+1, and 16*n*+1 all primes. (all such numbers are Carmichael numbers)

1001 is the smallest super pseudoprime to base 10 that is not semiprime.

1001 is the smallest product of three distinct primes of the form 6n + 1.

1001 is equal to the average of the only known prime squares of the form n! + 1 (i.e. 21, X1, and 2E01).

1001 is the smallest number that is a (Fermat) pseudoprime simultaneously to bases 2, 3 and 5.

1001 is the smallest number which can be represented by a Loeschian quadratic form *a*^{2} + *ab* + *b*^{2} in four different ways with *a* and *b* positive integers. (the integer pairs (*a*,*b*) are (21,1E), (28,13), (31,8) and (34,3))

## 1002 to googol (10^{100}) (selected)Edit

1008 is the final value of *s* for the Lucas–Lehmer primality test for the smallest composite Mersenne number (1227).

1060 is the largest perfectly cyclic number.

1066 is the largest number n such that m^(n+1) == m (mod n) holds for all m.

1078 is the smallest *n*>0 such that 25*n*^{2}+1 is square.

10X0 is the largest idoneal number.

10XE is the smallest nonpalindromic number n such that n written backwards is multiple of n.

1107 is the smallest prime which is at the end of an arithmetic progression of 9 primes (147, 2X1, 437, 591, 727, 881, X17, E71, 1107).

1129 is the smallest Euler-Jacobi pseudoprime base 2 which is not Carmichael number.

1154 is the smallest number which is value of the Euler totient function but not of the Carmichael lambda function.

1185 is the smallest odd number *n*>1 such that *n*+2^{k} and *n*−2^{k} are both composite for all *k*≥1 such that 2^{k} < *n*

11XX is the only non-3-digit *n* with sum of cubes of digits of *n* equals *n*.

1227 is the smallest strong pseudoprime base 2.

123E is the smallest prime Friedman number.

1254 is the smallest base with larger conjectured smallest Sierpinski number and larger conjectured smallest Riesel number than base 1E4

1261 is the smallest prime which is at the end of an arithmetic progression of X primes (147, 2X1, 437, 591, 727, 881, X17, E71, 1107, 1261).

1270 is the smallest vampire number.

1320 is conjectured to be the only one doubly strictly absurd number (numbers >0 which can be written as m^n−m in two different ways).

1322 is the smallest number (and the only known even number) which is product of the primes in a Wieferich pair.

1328 is the smallest number which is the average number of a stapled interval. (this interval is {n, n+1, n+2, ..., n+k−1} for n=1320 and k=15, besides, no such numbers n exist for k<15) (a finite sequence of n consecutive positive integers is called "stapled" if each element in the sequence is not relatively prime to at least one other element in the sequence)

1334 is the smallest number which is the largest number of a stapled interval. (this interval is {n, n+1, n+2, ..., n+k−1} for n=1320 and k=15, besides, no such numbers n exist for k<15) (a finite sequence of n consecutive positive integers is called "stapled" if each element in the sequence is not relatively prime to at least one other element in the sequence)

133E is conjectured to be the largest Lucas-Fermat prime (prime of the form Lucas(2^n)).

1404 is the smallest number n such that n!×k±1 and n!÷k±1 are all composite for all 1≤k≤n, where ! is the factorial.

1420 is the number of groups with order X8.

1486 is the largest known value of n achieving records in term in Recamán's sequence where n appears for first time (appears in term 38262519330314, the next such number is likely 351527, but this number does not appear in the first 10^{100} terms of Recamán's sequence).

1560 is the largest number such that no number less than its twice has more divisors.

1563 is the smallest n≥1 such that 3×36^{n}−1 is prime. (note that there is prime of the form 3×k^{n}−1 with n≤200 for all other even k<300, the largest n for these k is 1E6 (for k=158))

1589 is the smallest n>1 such that 31×2^{n}−1 is prime.

15X6 is the smallest *n*>0 such that 4X*n*^{2}+1 is square.

1677 is the largest imaginary quadratic field with class number 5.

1685 is the smallest generalized Wieferich prime base 10.

1691 is the smallest (Fermat) pseudoprime to both base 2 and base 3 that is not Carmichael number.

16E6 is the denominator of the first Bernoulli number which has an irregular pair (*B*_{10} = 497/16E6, which has an irregular pair (497, 10)).

1841 is the smallest k such that 3^k = 2 (mod k).

18X3 is conjectured to be the only number to appear 7 or more times in Pascal's triangle.

1961 is conjectured to be the largest panconsummate number.

19E3 is the smallest *n* such that if Dickson's conjecture holds, then there is an interval of *n* consecutive integers which has > π(*n*) primes (where π is the prime-counting function).

1X93 is the smallest composite Wieferich number (the first definition). (for the second definition, the such number is 19)

1E03 is the largest odd negated discriminants of orders of imaginary quadratic fields with 1 class per genus.

2047 is the largest known Wieferich prime. (the only other known such prime is 771)

20E0 is the smallest *n*>0 such that 3X*n*^{2}+1 is square.

2217 is the largest imaginary quadratic field with class number 6.

2440 is the largest number n such that no x^4 mod n is prime.

2453 is the largest Ramanujan-Nagell triangular number.

2455 is the smallest Euler-Jacobi pseudoprime (also the smallest strong pseudoprime) base 920.

2467 is the smallest prime with 10 the smallest primitive root.

2497 is the smallest cyclic number.

2505 is the smallest composite Fibonacci number with prime index.

2528 is conjectured to be the largest number which is not the sum of two t-primes, where a t-prime is a prime which has a twin.

2581 is the smallest non-palindromic balanced prime.

2600 is the smallest 7-hemiperfect number.

2631 is the largest known value of *p* such that ((*p*+1)^{p}−1)/(*p*^{2}) is (probable) prime.

2642 is conjectured to be the only *n* such that both *n*−1 and *n*+1 are (Fermat) pseudoprimes base 2.

2742 is the magic constant of the smallest magic square using 1 and consecutive prime numbers start from 3. (note that magic square using only prime numbers cannot contain the prime 2)

299E is the smallest prime which is the larger of a Wieferich pair with both primes are odd primes. (this Wieferich pair is {6E, 299E})

2X04 is the only square number besides 0 and 1 which is also a square pyramidal number.

2E00 is conjectured to be the largest number *n* such that σ(*n*) ≥ *e*^{γ}*n* ln(ln(*n*)).

2E01 is the largest square number of the form *n*!+1 (related to the Brocard's problem, n!+1 = m^2 has only three nonnegative integer solutions: (n,m) = (4,5), (5,E), and (7,5E)).

3003 is the only known n such that phi(n−1) = phi(n) = phi(n+1) (where phi is Euler totient function).

31XE is the smallest strong Lucas pseudoprime.

31E0 is the largest discriminant of the imaginary quadratic fields with 1 class per genus.

3202 is the smallest composite even *n* not divisible by 5 such that Fibonacci(*n*) = (*n*|5) (mod *n*), where (n|5) is the Legendre symbol.

3413 is the smallest n such that n and n+1 are both abundant.

3415 is the smallest odd number which is not of the form 2*n*^{2}+*p* with *p* prime.

3416 is the largest Lucas number which is also a triangular number.

3503 is the smallest Catalan pseudoprime. (the only two other known Catalan pseudoprimes are the squares of the two known Wieferich primes)

3553 is the smallest *n*>0 such that 57*n*^{2}+1 is square.

3557 is the smallest composite Wilson number.

3575 is conjectured to be the largest odd number which is not of the form 2*n*^{2}+*p* with *p* prime.

35E7 is the largest prime in which all substrings are also primes.

3849 is the smallest n such that 27E×2^{n}+1 is prime.

3916 is the smallest k>3 such that k−1 and k+1 have the same deficiency.

3953 is the index of the largest known Fermat prime.

3X06 is the smallest *n*>0 such that 64*n*^{2}+1 is square.

4170 is the largest triangular number which is also a tetrahedral number.

4200 is the largest powerful number which is also highly abundant.

4340 is the largest negated discriminants of orders of imaginary quadratic fields with 1 class per genus.

4401 is the smallest prime factor of googol+1.

444E is the largest known value of *p* such that (*p*^{p}−1)/(*p*−1) is (probable) prime.

4451 is conjectured to be the largest prime number which is not the sum of another prime number and a nonzero square.

4460 is the smallest number which is highly composite but not superabundant.

4700 is the smallest number which is order of sporadic simple group.

47X2 is conjectured to be the largest number that cannot be written as a sum of 6 or fewer cubes.

48X7 is conjectured to be the largest number which is a repunit in three or more bases (not including base 1).

4901 is the smallest non-Fermat prime p such that (4^(p-1)-1) == 0 mod ((p-1)^2+1).

4X95 is the sum of prime factors of the smallest Ruth–Aaron triplet (definition 1).

4E87 is the largest known value of *p* such that ((*p*−1)^{p}−1)/(*p*−2) is (probable) prime.

5147 is the smallest n such that k^n − (k−1)^n is (probable) prime, where k is 100

5324 is the smallest *n*>0 such that 45*n*^{2}+1 is square.

547E is the smallest (p, p−E) irregular prime.

56XX is conjectured to be the largest nonsquare even number which is not the sum of a prime number and a square (including 0).

5809 is the smallest number >1 which is both square and pentagonal (5-gonal).

5X1E is the smallest positive base such that there are no primary pretenders < the smallest Carmichael number (3X9).

6300 is the smallest n>1 such that x^y = y^(nx) has ≥ (d(n) + 4) solutions (where d(n) is the number of divisors of n).

6661 is the smallest number not of the form p*q, p and q prime with q=2*p-1, such that exactly half of the a such that 0<a<n and (a,n)=1 satisfy a^(n-1) = 1 (mod n). (all numbers of the form p*q, p and q prime with q=2*p-1, satisfy that exactly half of the a such that 0<a<n and (a,n)=1 satisfy a^(n-1) = 1 (mod n)) (note that this number is the smallest candidate of beastly prime, i.e. beastly numbers (numbers contain 666) end with 1, 5, 7 or E (digits coprime to 10))

6667 is the smallest beastly prime.

6999 is the smallest number >1 which is both hexagonal (6-gonal) and octagonal (8-gonal).

7294 is the smallest number which is a member of sociable numbers (length > 2).

7867 is the sum of all idoneal numbers.

7E94 is the largest number that cannot be written as a sum of 14 or fewer fourth powers. (all numbers of the form 27×14^{n} cannot be written as a sum of 13 or fewer fourth powers)

8327 × 17*d* is 6-digit repdigit *dddddd* for nonzero digit *d*.

8350 is the smallest number of the only known sociable numbers with length >9 (24).

866E is the only known nontrivial Wieferich-non-Wilson prime. (2 and 3 are trivial Wieferich-non-Wilson primes) (note that this number is the 1000th non-Wilson prime)

8833 is the smallest odd squarefree abundant number.

9316 is the smallest n≥1 such that 3×300^{n}−1 is prime.

9375 is the largest prime in the smallest set of the primes of the form *nk*+*x*, where *x* runs through the set of the integers 0≤*x*≤*n*−1 and gcd(*x*,*n*)=1, are all primes, for *n* = 16, the largest number *n* such that the set of the integers 0≤*x*≤*n*−1 and gcd(*x*,*n*)=1, when mod any prime *p*, is not the complete set {0, 1, 2, ..., *p*−1} (the corresponding *k* is 624).

98E7 is the smallest Wolstenholme prime.

9EX5 is conjectured to be the largest n such that there is no k≤sqrt(n)*ln(n) such that n*k±1 are twin primes.

E414 is the largest square number which is also a tetrahedral number.

E555 is conjectured to be the largest non-repunit permutable prime.

E800 is the smallest order such that there are two non-isomorphic (non-cyclic) simple groups.

E801 is the largest number which is not the sum of two abundant numbers.

EEEE is the only known repdigit number which is a member of betrothed numbers.

10667 is conjectured to be the largest nonsquare number which is not the sum of a prime number and a square (including 0).

10765 is the smallest positive base (also the smallest negative base) such that there are no Euler primary pretenders < the smallest absolute-Euler pseudoprime (1001).

11031 is the smallest happy prime.

116X3 is the odd k divisible by 3 < the smallest odd k divisible by 3 such that k×2^n+1 is composite for all even n (32759) require the largest n (1303EE8) to make k×2^n+1 to be prime.

117XX is the smallest negative base such that there are no primary pretenders < the smallest Carmichael number (3X9).

12345 is the smallest Smarandache prime.

13665 is the smallest prime *p* such that the number of primes end with 1 or 5 ≤ *p* is more than the number of primes end with 3, 7 or E ≤ *p*. (of course, 3 is the only prime ends with 3)

13885 is the smallest non-primepower *k* such that binomial(2*k*−1, *k*−1) = 1 (mod *k*).

14060 is the smallest number divisible by all natural numbers from 1 to 10.

14X28 is the smallest number not ends with 0 which is possible order of nonsolvable group.

15467 is the smallest composite Euclid number.

15600 is the smallest 4-perfect number.

15E30 is the smallest *n*>0 such that 71*n*^{2}+1 is square.

16661 is the smallest palindromic square with a non-palindromic square root.

16959 is conjectured to be the smallest number *n* divisible by 3 such that *n*×2^{k}+1 and *n*×2^{k+1}+1 are not both primes for all *k*≥1. (currently, the smallest such *n* with unknown status is 43)

16E61 is the largest square number of the form 2^{n}−7 (related to the Ramanujan–Nagell equation, 2^n−7 = m^2 has only five nonnegative integer solutions: (n,m) = (3,1), (4,3), (5,5), (7,E), and (13,131)).

16E68 is the smallest number >1 which can be written as *k*^{p} with *p* odd prime in 2 different ways.

19694 is the smallest number which is the sum of the numbers in a stapled interval.

1X537 × 7*d* is 6-digit repdigit *dddddd* for nonzero digit *d*.

1X795 is the smallest number *n*>3 such that 2^{n}−*n*−2 is (probable) prime.

1X8E4 is the largest Dudeney number.

1E143 is conjectured to be the smallest nonsquare k divisible by 3 such that k×2^n−1 is composite for all even n≥2.

1E397 is the odd k < the smallest odd k such that 2^n+k is composite for all n (39565) require the largest n (3065974) to make 2^n+k to be (probable) prime.

1E400 is the smallest factorial (8!) which is not highly composite number.

1E51E is the smallest odd non-generous prime.

1E703 is the smallest number >1 which is both pentagonal (5-gonal) and hexagonal (6-gonal).

1E901 is the smallest Carmichael number which is not sphenic number.

20EX5 is conjectured to be the largest number *n* such that *n*+*k*^{2} is composite for all *k* such that *k*^{2} ≤ *n* (the conjectured largest nonsquare number *n* such that *n*−*k*^{2} is composite for all *k* such that *k*^{2} ≤ *n* is 10667)

23000 is conjectured to be the largest number that can be written as a sum of a perfect power (including 1, but not including 0) and a prime in a unique way.

26808 is the smallest *n*>0 such that 75*n*^{2}+1 is square.

26E25 is a value of *n* such that *n*/ln(*n*) is near-integer (i.e. 26E25/ln(26E25) = 2X12.000003 78XX35 44729E 7745E6 ...). (another such value of *n* is 117, 117/ln(117) = 27.EEEEE8 296X76 X3E292 886458 ...)

2718E is the smallest prime p such that p#×q±1 and p#÷q±1 are all composite for all primes 2≤q≤p, where # is the primorial.

2765E is the smallest prime *p* such that *p*^{2} divides *n*! − (*n*−1)! + (*n*−2)! − ... 1! for some *n* > 3 (*n* = 13439).

2E69E is the smallest prime *p* such that the number of primes end with 1 or E ≤ *p* is more than the number of primes end with 5 or 7 ≤ *p*.

31E00 is the largest number *n* such that carmichael_lambda(*n*) = 10.

31E14 is the value of 2[5]3 (where *a*[*n*]*b* is the hyperoperation).

31E15 is conjectured to be the largest Fermat prime.

32759 is the smallest k divisible by 3 such that k×2^n+1 is composite for all even n≥2.

39565 is conjectured to be the smallest Sierpinski number (numbers n such that n*2^k + 1 is composite for all k ≥ 1). (currently, the smallest n with unknown status is 10311)

3X733 is the smallest odd unitary admirable number.

3E554 is the only known number >1 that can be written in bases 2 to 5 using only the digits 0 and 1.

41414 is the largest undulating square.

45023 is conjectured to be the only odd number k besides 3 such that k−1 and k+1 have the same deficiency.

47183 is conjectured to be the smallest k divisible by 3 such that k×2^n+1 is composite for all odd n≥1.

49872 is term in Recamán's sequence where 17 appears. (all other numbers ≤50 appear the first 100 terms of Recamán's sequence)

511E7 is the largest two-sided prime.

5E685 is the largest known generalized Wieferich prime base 10. (the only other known such prime is 1685)

63X00 is the largest number *k* such that for any positive integers *x*, *y* coprime to *k*, *x*^{x} = *y* (mod *k*) if and only if *y*^{y} = *x* (mod *k*).

78298 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 33 with 0 <= |x| <= |y| <= |z|.

7923X is the smallest even weak pseudoprime base 2.

7X42E is the magic constant of the smallest magic square using consecutive prime numbers start from 3. (note that magic square using only prime numbers cannot contain the prime 2)

83E19 is conjectured to be the smallest k divisible by 3 such that k×2^n−1 is composite for all odd n≥1.

8914E is term in Recamán's sequence where 64 appears.

89159 is term in Recamán's sequence where 51 appears.

95900 is the kissing number in 20 dimensions. (this is the Leech lattice)

X0693 is the largest triangular number which is also a square pyramidal number.

100151 is the smallest prime which is at the end of an arithmetic progression of E primes.

108181 is the smallest prime which is at the end of an arithmetic progression of 10 primes.

10X620 is the smallest *n*>0 such that 61*n*^{2}+1 is square.

110XX1 is conjectured to be the smallest prime Sierpinski number.

111101 is the smallest Perrin pseudoprime.

111111 is the smallest repunit number which is not sum of two square numbers.

119001 is the smallest non-Fermat prime p such that p = 2*phi(sigma((p-1)/2))+ 1.

122461 is the smallest super-Poulet number which is not semiprime.

146EE7 is conjectured to be the largest number that cannot be written as a sum of 4 or fewer tetrahedral numbers.

156000 is the smallest factorial (9!) which is not highly abundant number.

160061 is the smallest palindromic square with an even number of digits.

172X55 is the smallest composite k such that gcd(k^2, 2^(k-1) - 1) > k.

175E71 is the smallest odd number which is product of the primes in a Wieferich pair.

1814E7 is the smallest n such that (n−1, n, n+1) is Ruth–Aaron triplet (definition 2). (the only other known such n is 140E570705)

186X35 is the largest cyclic number not starting with 0.

206817 is conjectured to be the smallest Riesel number (numbers n such that n*2^k − 1 is composite for all k ≥ 1). (currently, the smallest n with unknown status is 13E1)

208699 is the smallest n such that both n−1 and n+1 are weird numbers.

220990 is the smallest number n>10 such that n±1, n/2±1, n/3±1 are all primes.

2X5626 is the largest value of the smallest n such that k×10^n+1 is prime for some k which is not power of 10 < the smallest k which is not power of 10 such that k×10^n+1 is composite for all n (375) (k=298).

351527 is the smallest number which is not known to appear in Recamán's sequence.

37761E is the smallest Perrin pseudoprime which is not prime power.

3966E9 is the smallest number whose square is pandigital.

406217 is the largest known Mirimanoff prime (generalized Wieferich prime base 3). (the only other known such prime is E)

406440 is the smallest n such that n±E, n±11, n±15, n±17 are all primes (two consecutive prime quadruplets).

460089 is the smallest n≥1 such that 4×60^{n}−1 is prime.

50E8E8 is the smallest number which belong to sociable numbers with length 4. (it is conjectured that there are no sociable numbers with length 3)

522E58 is conjectured to be the largest number that cannot be written as a sum of 5 or fewer cubes.

56057E is the Skewes number of twin primes.

562E31 is the smallest number which is a strong pseudoprime to both base 2 and base 3.

582992 is the largest tetrahedral number which is also a pronic number.

624451 is the smallest *N* such that the multiplicative group of integers modulo *N* ([*Z*/*NZ*]^{×}) as a product of cyclic groups *C*_{k1} × *C*_{k2} × ... × *C*_{kn} contains a copy of *C*_{n} for some even number *n* (*n*=122) and has *m*>2 (*m*=3, the corresponding group is *C*_{2} × *C*_{122} × *C*_{2440}).

715261 is conjectured to be the largest number which is not the sum of a perfect power (including 1, but not including 0) and a prime.

734X05 is the smallest composite generalized Wieferich number base 10.

7X0X01 is the smallest 4-hyperperfect number.

865687 is the largest known Wolstenholme prime. (the only other known such prime is 98E7)

E77115 is conjectured to be the largest non-repunit circular prime.

EX3711 is the largest number whose square contains no digit more than once.

1111111 is the smallest composite repunit with prime length.

118X582 is the smallest n such that sigma_5(n) is prime.

1262827 is the smallest prime *p* which divides (*p*−1)! − (*p*−2)! + (*p*−3)! − ... 1!.

1292300 is the number of possible moves for a 2×2×2 Rubik's Cube.

1303EE8 is the largest value of the smallest even n such that k×2^n+1 is prime for some odd k divisible by 3 < the smallest odd k divisible by 3 such that k×2^n+1 is composite for all even n (32759) (k=116X3).

18E1271 is the Skewes number of cousin primes. (the Skewes number of sexy primes is still unknown)

22128X9 is the smallest odd number >5 not of the form p + 2^a + 2^b, a, b > 0, p prime.

2357E11 is the largest known Smarandache–Wellin prime. (the only other known such prime is 2)

2443901 is the smallest pseudoprime Chebyshev number.

247983E is the lesser of the smallest Ruth-Aaron number pairs in both of two definitions such that both of the two numbers are not squarefree.

2X3X00X is the smallest even Fibonacci pseudoprime.

2E98800 is the smallest 9-hemiperfect number.

3065974 is the largest value of the smallest n such that 2^n+k is (probable) prime for some odd k < the smallest odd k such that 2^n+k is composite for all n (39565) (k=1E397).

5653662 is the smallest even Perrin pseudoprime.

6E8XE77 is the smallest weakly prime.

8594301 is the smallest number which is a strong pseudoprime to bases 2, 3 and 5 simultaneously.

9321341 is the smallest restricted Perrin pseudoprime.

X32X735 is conjectured to be the smallest odd integer neither of the form p + 2^k nor of the form p − 2^k with k > 0, and p prime.

E2X20X8 is the smallest power of 2 starts with E.

24009593 is the smallest number *n* which is not two less than Fermat number and eulerphi(*n*) = (*n*+1)/2.

24727225 is conjectured to be the only unique prime not of the form Φ_{n}(10), where Φ is the cyclotomic polynomial.

25646000 is the number of possible positions in a 2×2×2 Rubik's cube reachable from the starting position. (this number is exactly the "possible moves" of Rubik's cube iff the order is odd, and 20 times the "possible moves" of Rubik's cube iff the order is even)

25E63007 is the smallest n such that (n−1, n, n+1) is Ruth–Aaron triplet (definition 1). (the only other known such n is 253692112X4)

2696E854 is the smallest friend of 20.

49363E53 is the smallest n such that n and n±1 are all abundant.

4X089969 is the smallest odd abundant-perfect number.

4EE2308E is the smallest composite Wagstaff number.

638X4190 is the smallest *n*>0 such that 51*n*^{2}+1 is square.

9X64EEEE is the smallest prime *p* such that *f*(*p*) ≠ *f*(*p*−1) + 1, where *f*(*n*) is minimal number of 1's required to build *n* using "+" and "×".

XX000001 is the largest minimal prime.

E187EX26 is the largest known Wilson number.

E9352176 is the largest Lynch-Bell number.

EX1E1400 is the number of divisors of X7X33098E169885E2331606809545400000000000000000000 (the order of Monster simple group).

107X71595 is the smallest non-Fermat prime p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

120670630 is conjectured to be the largest perfect power (10E96^{2}) such that there are no primes between it and the next larger perfect power (120670647 = 47^{5}).

197E21947 is the smallest 3-powerful number which can be written as sum of two positive 3-powerful numbers (67E97X7 + 191324160 = 1X7^{3} + 2^{3}×3^{5}×61^{3} = 647^{3} = 197E21947). (the next such number is 489EE7E74454 = 2^{7}×5^{4}×255^{3})

213574615 is the smallest *n* > 1 such that (−1)^{Ω(1)} + (−1)^{Ω(2)} + ... + (−1)^{Ω(n)} > 0 (where Ω(*k*) is number of primes dividing *k* (counted with multiplicity)). (i.e. this number is the smallest counterexample to Polya's conjecture)

213574616 is the smallest *n* such that (−1)^{Ω(1)} + (−1)^{Ω(2)} + ... + (−1)^{Ω(n)} > 1 (where Ω(*k*) is number of primes dividing *k* (counted with multiplicity)).

237E3745E is the smallest non-Fermat prime p such that phi(p-2) = phi(p-1) and simultaneously Product_{d|(p-2)} phi(d) = Product_{d|(p-1)} phi(d), where phi(x) = Euler totient function.

285985700 is the smallest number which is superabundant but not highly composite.

4EE2308X7 is the maximum positive value for a 28-bit signed binary integer in computing.

51E746E84 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 26 with 0 <= |x| <= |y| <= |z|.

69E63848E × 1E*d* is E-digit repdigit *ddddddddddd* for nonzero digit *d*.

75885X947 is the smallest number which is a strong pseudoprime to bases 2, 3, 5 and 7 simultaneously.

963848369 is the smallest palindromic square >9 beginning and ending with 9.

9EX461593 is conjectured to be the largest odd number which is number of sides of a constructible polygon.

9EX461595 is the smallest composite Fermat number.

X3XE85429 is the smallest magic constant for any 3×3 magic square made from consecutive primes.

XE20601X1 is the smallest *n* such that 2^{n} = 3 mod *n*.

10656X5641 is the smallest abundant number coprime to the smallest abundant number (10).

123456789E × E*d* is E-digit repdigit *ddddddddddd* for nonzero digit *d*.

2859857000 is the smallest superior highly composite number which is not colossally abundant number.

28E9806000 is the smallest 5-perfect number.

302E11E52E is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 20 with 0 <= |x| <= |y| <= |z|.

3577777799 is the smallest number with multiplicative persistence 7.

375EE5E515 is the largest right-truncatable prime.

3X67X54832 is the only Münchausen number beside 1.

5907632598 is the smallest n such that both n and n+1 are admirable numbers.

7843E471E6 is the smallest n such that n±1, n±E, n±11, n±15, n±17 are all primes (Hargrave primes or quintuplet twin Primes).

1000195E341 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 44 with 0 <= |x| <= |y| <= |z|.

22777E33855 is conjectured to be the largest prime p such that p−1 is perfect number.

27169742600 is the smallest colossally abundant number which is not superior highly composite number.

9X03693X831 is the smallest prime *p* such that the number of primes end with 1 or 7 ≤ *p* is more than the number of primes end with 2, 5 or E ≤ *p* (of course, 2 is the only prime ends with 2).

EX987643205 is the largest number which does not contain any digit more than once and which is not divisible by any of its digits.

EX987654021 is the largest prime with distinct digits.

1023456789XE is the smallest pandigital number.

124X7E538609 is the smallest pandigital square.

2X9169X76EE7 is the smallest number which is a strong pseudoprime to all prime bases ≤10.

3E42E7303437 is conjectured to be the largest double Wagstaff prime.

412118031327 is the only known n not end with 2, 3, 8, 9 such that phi(n) = phi(n+1) (where phi is Euler totient function).

502388973590 is the smallest *n*>0 such that 111*n*^{2}+1 is square.

821000001000 is the only one autobiographical number.

9E0E77984EX2 is conjectured to be the largest number that cannot be written as a sum of 4 or fewer cubes. (all numbers = 4 or 5 mod 9 cannot be written as a sum of 3 or fewer cubes)

X31X85203623 is the value of 3[5]2 (where *a*[*n*]*b* is the hyperoperation).

E8750X649321 is the largest square with distinct digits.

101234568X79E is the smallest pandigital prime.

1846395623990 is the smallest *n*>0 such that 91*n*^{2}+1 is square.

1X33033599509 is the largest known Wieferich number.

19E39038119000 is conjectured to be the largest highly composite numbers whose number of divisors is also a highly composite number.

38262519330314 is the largest known record in term in Recamán's sequence where n appears for first time (for n=1486).

41888777397801 is the smallest order 2 Carmichael number.

8XE305887E4937 is the largest known prime Motzkin number. (the only other three known prime Motzkin numbers are 2, X7, and 8E87)

92X79E43715865 × 15*d* is 14-digit repdigit *dddddddddddd* for nonzero digit *d*.

6XX48E23X989748 is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 29 with 0 <= |x| <= |y| <= |z|.

5288E5719467971X is the smallest value of |z| of the solution to x^3 + y^3 + z^3 = 36 with 0 <= |x| <= |y| <= |z|.

110X9695X887X2X75 is conjectured to be the smallest *n* which is not power of 2 and not congruent to 1 mod 3 (i.e. not end with 1, 4, 7 or X) (in which all such numbers are divisible by 3) such that 2×*n*^{k}+1 is composite for all *k* ≥ 1. (currently, the smallest such *n* with unknown status is 265)

15059399520368520 is the near-integer of the Ramanujan's constant.

18833100X792X68X77 is the smallest number which is a strong pseudoprime to all prime bases ≤20.

327EX58X0416159433 is conjectured to be the smallest Riesel number divisible by 3.

982571952E134086X9 is conjectured to be the smallest Sierpinski number divisible by 3.

9EX461594000000000 is the number of divisors of 279742032253400191X3732493936E5325816940277445X69E6E640709X213E081341121859588575205X7453064XX41E539X230865412630 (the period of the sequence of the primary pretender to base *n*).

175E360000000000000 is the number of possible moves for a 3×3×3 Rubik's Cube (also the number of possible positions in a 3×3×3 Rubik's cube reachable from the starting position)

5969X55302611400000 is the smallest 6-perfect number.

X47065X6E31E5070X311 is conjectured to be the smallest Brier number (numbers n such that n*2^k + 1 and n*2^k − 1 are both composite for all k ≥ 1).

322160E168057572552280 is the largest known unitary perfect number.

5420X1E50396995X663X4246020 is the largest known number which appears ≥5 times in Pascal's triangle.

606890346850EX6800E036206464 is the largest polydivisible number.

471X34X164259EX16E324XE8X32E7817 is the largest left-truncatable prime.

2X695925806818735399X37X20X31E3534X7 is conjectured to be the largest double Mersenne prime.

X7X33098E169885E2331606809545400000000000000000000 is the order of Monster simple group.

15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E is the largest narcissistic number.

18E27099E934907277099E91X484159E264E0XX1359X268567863258910E7498X682454 is the largest known sublime number.

59E18922E81631X39875663E89X853X91E595336X6114815X5X6929933X288E774E479575X628 is conjectured to be the largest power of 2 not contain the digit 0.

279742032253400191X3732493936E5325816940277445X69E6E640709X213E081341121859588575205X7453064XX41E539X230865412630 is the period of the sequence of the primary pretender to base *n*.

3983226032906342E525149932E15661656639379695969X0993370X82917XEX261716E617X3366743887X5X49X523X70816109X87683E82357E17683XXE63E32587348E4900000 is the largest superabundant number which is also highly composite number.

7795E6X68X4315X011E278956400624X6E089641887X743601896X6218521E45895475XX143988649460888E46407E9X123E64624E8435E78X25485E165E122007E787872584194 is the value of 4→3→2 (where “→” is the Conway chained arrow notation).

## The two surprises... the two extra number > googol... with 172 digits and 180 digits!!!Edit

157EE3E44011433003856565X452XX65EX32E4761X98909510037975834264E94XE45353X8E10E3507E1171X3824EX557X840684708568556011E07X1E774806E765X5751X83X624077258X75957E520XE84X24351927E5817X1X5195E89E703449401541587283846X5155814492X4X644883 is the period of the sequence of the Euler primary pretender to base *n*.

16E3X62E27E84094585333XX1E269E2E5679E9110XX37444E957178X819EE685789X705764610X4XX40E9E03750XE99X212X5X658628725574645E4E97X389623109E1EE435X2EE0611X32872211895X0461E8889414E725EX14E36X53794473333X6X5155E26545233424264X968272273E5E0166027782 is conjectured to be the largest number *n* such that *n*−1, *n*, and *n*+1 all have primitive roots. (the only other known such numbers *n* are 2, 3, 4, 5, 6, X, 16, 22, 6X, 182)

## Sequence of uninteresting numbersEdit

Numbers that are not (primes, 10-smooth, perfect powers, or palindromes):

2X, 32, 3X, 43, 49, 4X, 52, 58, 59, 62, 64, 6X, 71, 72, 73, 78, 79, 7X, 7E, 86, 8X, 93, 96, 97, 98, 9X, 9E, X2, X3, X4, X9, E1, E2, E4, E6, E9, EX, 102, 104, 108, 109, 10E, 110, 112, 113, 115, 118, 11X, 122, 123, 124, 126, 129, 12X, 132, 133, 134, 135, 136, 137, 138, 13X, 142, 143, 149, 14X, 14E, 150, 152, 153, 154, 155, 158, 159, 15X, 15E, 162, 163, 165, 166, 16X, 170, 172, 174, 176, 177, 178, 179, 17X, 184, 186, 187, 188, 189, 192, 193, 196, 197, 198, 199, 19X, 1X2, 1X3, 1X4, 1X8, 1X9, 1XX, 1E0, 1E2, 1E3, 1E6, 1E8, 1E9, 1EX, 1EE, 203, 204, 207, 208, 20X, 20E, 211, 213, 214, 215, 216, 219, 21X, 220, 224, 226, 227, 229, 22X, 22E, 231, 233, 234, 235, 238, 239, 23X, 23E, 243, 244, 245, 246, 248, 249, 24X, 250, 253, 256, 257, 258, 259, 25X, 264, 265, 266, 268, 269, 26X, 26E, 270, 274, 275, 278, 279, 27X, 283, 284, 286, 287, 289, 28X, 28E, 293, 296, 297, 298, 29X, 29E, 2X0, 2X3, 2X4, 2X5, 2X6, 2X7, 2X8, 2X9, 2XX, 2E3, 2E4, 2E5, 2E6, 2E7, 2E8, 2E9, 2EX, 302, 304, 305, 306, 30X, 310, 311, 312, 317, 318, 319, 31X, 31E, 320, 322, 324, 328, 329, 32X, 330, 331, 332, 334, 335, 336, 337, 338, 339, 33X, 341, 342, 345, 348, 349, 350, 351, 352, 354, 355, 356, 359, 35X, 361, 362, 364, 366, 367, 369, 36X, 36E, 370, 371, 372, 374, 376, 378, 37X, 37E, 382, 384, 385, 386, 387, 388, 389, 38X, 392, 394, 395, 396, 398, 399, 39E, 3X0, 3X1, 3X2, 3X4, 3X6, 3X7, 3X9, 3XX, 3E0, 3E1, 3E2, 3E4, 3E6, 3E8, 3E9, 3EX, 3EE, 402, 403, 405, 406, 407, 408, 409, 40X, 411, 412, 413, 417, 418, 419, 41X, 422, 423, 426, 428, 429, 42X, 42E, 430, 432, 433, 436, 438, 439, 43X, 43E, 440, 442, 443, 445, 448, 449, 44X, 44E, 450, 451, 452, 453, 456, 458, 459, 45X, 461, 462, 463, 466, 467, 468, 469, 46X, 472, 473, 475, 476, 477, 478, 479, 47X, 47E, 482, 486, 487, 488, 489, 48X, 490, 491, 493, 495, 496, 498, 49X, 49E, 4X0, 4X1, 4X2, 4X3, 4X6, 4X7, 4X9, 4XX, 4XE, 4E0, 4E2, 4E3, 4E5, 4E6, 4E7, 4E8, 4E9, 4EX, 501, 502, 503, 504, 508, 50X, 50E, 510, 512, 514, 516, 518, 519, 51X, 520, 521, 522, 523, 524, 528, 529, 52X, 52E, 532, 533, 534, 536, 537, 538, 539, 53X, 53E, 543, 544, 546, 547, 548, 549, 54X, 54E, 550, 551, 552, 553, 556, 558, 559, 55X, 55E, 561, 562, 563, 564, 566, 567, 569, 56X, 56E, 570, 571, 572, 573, 574, 578, 579, 57X, 57E, 580, 581, 582, 583, 584, 586, 588, 58X, 590, 592, 593, 594, 596, 597, 598, 599, 59X, 5X2, 5X3, 5X4, 5X6, 5X8, 5X9, 5XX, 5XE, 5E0, 5E2, 5E3, 5E4, 5E6, 5E8, 5E9, 5EX, 601, 602, 603, 604, 605, 607, 608, 609, 60X, 610, 612, 613, 618, 619, 61X, 620, 621, 622, 624, 625, 627, 629, 62X, 62E, 631, 632, 633, 634, 635, 638, 639, 63X, 640, 641, 642, 643, 644, 645, 648, 649, 64X, 64E, 651, 652, 653, 654, 657, 658, 659, 65X, 65E, 660, 662, 663, 664, 667, 668, 66X, 670, 671, 672, 673, 674, 677, 678, 679, 67X, 67E, 682, 683, 684, 685, 689, 68X, 691, 692, 693, 694, 697, 699, 69X, 6X0, 6X1, 6X2, 6X3, 6X4, 6X5, 6X8, 6X9, 6XX, 6XE, 6E0, 6E2, 6E3, 6E5, 6E7, 6E8, 6E9, 6EX, 6EE, 702, 703, 704, 706, 708, 709, 70X, 710, 712, 713, 715, 716, 718, 71X, 720, 722, 723, 724, 725, 726, 728, 729, 72X, 72E, 730, 731, 732, 733, 734, 738, 739, 73X, 73E, 741, 742, 743, 744, 746, 748, 749, 74X, 74E, 750, 752, 753, 754, 755, 756, 758, 759, 75E, 761, 762, 763, 764, 765, 766, 768, 76X, 770, 772, 773, 774, 776, 779, 77X, 780, 781, 782, 783, 784, 786, 788, 789, 78X, 78E, 790, 792, 793, 795, 796, 798, 79X, 79E, 7X0, 7X2, 7X3, 7X4, 7X5, 7X8, 7X9, 7XX, 7XE, 7E0, 7E1, 7E2, 7E3, 7E4, 7E5, 7E6, 7E8, 7E9, 7EX, 802, 805, 806, 807, 809, 80X, 810, 811, 812, 813, 814, 815, 816, 819, 81X, 81E, 821, 822, 823, 824, 826, 827, 829, 82X, 831, 832, 833, 834, 836, 837, 839, 83X, 83E, 842, 843, 844, 845, 846, 847, 849, 84E, 850, 852, 854, 856, 857, 859, 85X, 860, 862, 863, 864, 866, 869, 86X, 86E, 870, 872, 873, 874, 875, 876, 877, 879, 87X, 87E, 880, 883, 884, 885, 886, 887, 889, 88X, 891, 892, 893, 894, 895, 896, 897, 899, 89X, 89E, 8X0, 8X1, 8X2, 8X3, 8X4, 8X6, 8X9, 8XX, 8E0, 8E1, 8E2, 8E3, 8E4, 8E6, 8E9, 8EX, 8EE, 902, 903, 904, 906, 908, 90X, 910, 911, 912, 913, 914, 915, 916, 917, 918, 91X, 922, 924, 925, 926, 928, 92X, 930, 931, 932, 933, 934, 935, 936, 937, 938, 93X, 93E, 941, 942, 943, 944, 945, 947, 948, 94X, 94E, 950, 951, 952, 953, 954, 956, 957, 958, 95X, 960, 962, 963, 966, 968, 96X, 96E, 970, 972, 973, 974, 975, 977, 978, 97X, 97E, 980, 981, 982, 983, 984, 985, 986, 98X, 98E, 990, 991, 992, 993, 996, 997, 998, 99X, 99E, 9X0, 9X1, 9X2, 9X3, 9X4, 9X5, 9X6, 9X8, 9XX, 9E0, 9E2, 9E3, 9E4, 9E6, 9E7, 9E8, 9EX, X01, X02, X03, X05, X06, X08, X09, X12, X13, X14, X15, X18, X19, X1E, X20, X21, X22, X23, X24, X25, X28, X29, X2E, X30, X31, X32, X33, X34, X36, X38, X40, X42, X43, X44, X46, X47, X48, X49, X51, X52, X53, X54, X55, X56, X57, X58, X59, X61, X62, X63, X64, X65, X66, X67, X68, X70, X71, X72, X73, X74, X75, X76, X78, X79, X7E, X81, X82, X83, X85, X86, X88, X89, X8E, X90, X92, X93, X94, X96, X97, X98, X99, XX0, XX1, XX2, XX3, XX4, XX5, XX6, XX9, XE0, XE1, XE2, XE4, XE5, XE6, XE8, XE9, E01, E02, E03, E04, E05, E06, E07, E08, E09, E0X, E10, E12, E13, E16, E17, E18, E19, E1X, E20, E22, E23, E24, E26, E27, E28, E2X, E32, E33, E34, E35, E36, E38, E39, E3X, E40, E41, E42, E43, E44, E46, E47, E48, E49, E4X, E50, E51, E52, E53, E54, E55, E57, E58, E59, E5X, E60, E62, E63, E64, E65, E66, E68, E69, E6X, E70, E72, E73, E74, E75, E76, E77, E78, E79, E7X, E82, E83, E84, E85, E86, E87, E88, E89, E8X, E90, E93, E94, E96, E98, E9X, EX0, EX1, EX2, EX3, EX4, EX6, EX7, EX8, EX9, EXX, EE0, EE1, EE2, EE3, EE4, EE6, EE8, EE9, EEX, ...

The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural number is interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.