A Friedman number is an integer, which is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, and exponentiation. For example, 163 is Friedman number, since 163 = 3 × 61, besides, 445 is also Friedman number, since 445 = 5^{4} + 4.
Allowing parentheses without operators would result in trivial Friedman numbers such as 24 = (24). Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 001729 = 1700 + 29
There are 6E Friedman numbers ≤10000, the expressions of them are:
121 = 11^{2} | 127 = 21 × 7 | 135 = 31 × 5 | 144 = 41 × 4 | 163 = 61 × 3 |
346 = 6 × 3^{4} | 368 = 8^{6–3} | 376 = 73 × 6 | 441 = (4 + 1)^{4} | 445 = 5^{4} + 4 |
114X = 141 × X | 1169 = 161 × 9 | 1207 = 201 × 7 | 1228 = 2^{1+2+8} | 1229 = 2^{2+9} + 1 |
122X = 2 × (2^{X} + 1) | 122E = 2^{E} + 2 + 1 | 123E = 2^{E} + 13 | 1270 = 70 × 21 | 12E9 = 2^{E} + 91 |
1305 = 301 × 5 | 1323 = 3^{2×3+1} | 1350 = 50 × 31 | 1404 = 401 × 4 | 1428 = 814 × 2 |
1440 = 41 × 40 | 1476 = 7^{4} – 6 – 1 | 1477 = 7^{4} – 7 + 1 | 1481 = (8 – 1)^{4} × 1 | 1482 = 841 × 2 |
1544 = 4 × 1 × 5^{4} | 1603 = 601 × 3 | 1630 = 61 × 30 | 1826 = 81 × 26 | 1924 = 91 × 24 |
1X28 = X8 × 21 | 1E53 = (E + 5 – 1)^{3} | 2379 = 927 × 3 | 2448 = (4 × 2)^{4} – 8 | 2452 = 54^{2} – 2 |
2454 = 4^{5×2−4} | 2468 = 4^{6} + 8 × 2 | 2525 = 25^{2} × 5 | 2541 = (54 + 1)^{2} | 2545 = 55^{2} + 4 |
257X = 5 × 7 × X2 | 2636 = 6 × (3^{6} – 2) | 2779 = 79 × 7^{2} | 2815 = 58^{2} + 1 | 2942 = 2 × (9 – 2)^{4} |
2X84 = (X^{4} – 8) ÷ 2 | 2E36 = 326 × E | 3166 = 631 × 6 | 3266 = 63 × 62 | 3460 = (60 ÷ 4)^{3} |
3548 = 834 × 5 | 35X6 = 6 × (X^{3} + 5) | 3760 = 730 × 6 | 37E6 = 73E × 6 | 3963 = 3^{9} ÷ 3 – 6 |
3E76 = 7E3 × 6 | 416E = 461 × E | 45X8 = 548 × X | 46X8 = 6X4 × 8 | 47EX = E7 × 4X |
4892 = 2 × (8^{4} – 9) | 48X8 = (X – 2) × 8^{4} | 4969 = 649 × 9 | 5513 = 3 × 1 × 5^{5} | 5788 = 857 × 8 |
57EX = 75 × E × X | 5954 = (9 + 5 ÷ 5)^{4} | 597E = 9E5 × 7 | 5E14 = (E – 1) × 4^{5} | 5E22 = 2^{E} + 5 × 2 |
6946 = 9 × (6^{4} + 6) | 7651 = (5^{7} + 1) ÷ 6 | 7X28 = 8X × 2^{7} | 95X2 = 2^{9+5} + X | 9E2X = XE^{2} + 9 |
X0E2 = E0^{2} – X | X454 = 5X × 4^{4} | EX24 = X × 2^{E} – 4 |
A nice (or orderly) Friedman number is a Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, 346 = 3^{4} × 6. The first nice Friedman numbers are:
- 346, 1229, 122E, 1544, 1E53, 2448, 2525, 2942, 5513, ...
The smallest prime Friedman number is 123E = 2^{E} + 13
Michael Brand proved that the density of Friedman numbers among the natural numbers is 1, which is to say that the probability of a number chosen randomly and uniformly between 1 and n to be a Friedman number tends to 1 as n tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for base 2, base 3, and base 4 nice Friedman numbers. The case of base 10 nice Friedman numbers is still open (although 10 is the least common multiple of 2, 3, and 4).
Vampire numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1270 = 21 × 70.
It has been proven that repdigits of the greatest digit in any base with more than 20 digits are nice Friedman numbers (note that 20 is exactly double of 10).
Proof: Let
be the greatest digit of the repdigit in base
. Then
is a Friedman number. The number of digits in
is 21 by counting.