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 = 54 + 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 = 112 127 = 21 × 7 135 = 31 × 5 144 = 41 × 4 163 = 61 × 3 346 = 6 × 34 368 = 86–3 376 = 73 × 6 441 = (4 + 1)4 445 = 54 + 4 114X = 141 × X 1169 = 161 × 9 1207 = 201 × 7 1228 = 21+2+8 1229 = 22+9 + 1 122X = 2 × (2X + 1) 122E = 2E + 2 + 1 123E = 2E + 13 1270 = 70 × 21 12E9 = 2E + 91 1305 = 301 × 5 1323 = 32×3+1 1350 = 50 × 31 1404 = 401 × 4 1428 = 814 × 2 1440 = 41 × 40 1476 = 74 – 6 – 1 1477 = 74 – 7 + 1 1481 = (8 – 1)4 × 1 1482 = 841 × 2 1544 = 4 × 1 × 54 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 = 542 – 2 2454 = 45×2−4 2468 = 46 + 8 × 2 2525 = 252 × 5 2541 = (54 + 1)2 2545 = 552 + 4 257X = 5 × 7 × X2 2636 = 6 × (36 – 2) 2779 = 79 × 72 2815 = 582 + 1 2942 = 2 × (9 – 2)4 2X84 = (X4 – 8) ÷ 2 2E36 = 326 × E 3166 = 631 × 6 3266 = 63 × 62 3460 = (60 ÷ 4)3 3548 = 834 × 5 35X6 = 6 × (X3 + 5) 3760 = 730 × 6 37E6 = 73E × 6 3963 = 39 ÷ 3 – 6 3E76 = 7E3 × 6 416E = 461 × E 45X8 = 548 × X 46X8 = 6X4 × 8 47EX = E7 × 4X 4892 = 2 × (84 – 9) 48X8 = (X – 2) × 84 4969 = 649 × 9 5513 = 3 × 1 × 55 5788 = 857 × 8 57EX = 75 × E × X 5954 = (9 + 5 ÷ 5)4 597E = 9E5 × 7 5E14 = (E – 1) × 45 5E22 = 2E + 5 × 2 6946 = 9 × (64 + 6) 7651 = (57 + 1) ÷ 6 7X28 = 8X × 27 95X2 = 29+5 + X 9E2X = XE2 + 9 X0E2 = E02 – X X454 = 5X × 44 EX24 = X × 2E – 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 = 34 × 6. The first nice Friedman numbers are:

346, 1229, 122E, 1544, 1E53, 2448, 2525, 2942, 5513, ...

The smallest prime Friedman number is 123E = 2E + 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

${\displaystyle d = b - 1}$ be the greatest digit of the repdigit in base

${\displaystyle b}$ . Then

{\displaystyle \begin{align} d\sum_{i = 0}^{20}b^i & = \left(d + \frac{d}{d}\right)^{\left(\frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d}\right)\left(\frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d}\right)} - \frac{d}{d} \\ & = (d + 1)^{(5)(5)} - 1 \\ & = b^{21} - 1 = (b - 1)\sum_{i = 0}^{20}b^i = d\sum_{i = 0}^{20}b^i \end{align} }

is a Friedman number. The number of digits in

${\displaystyle \left(d + \frac{d}{d}\right)^{\left(\frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d}\right)\left(\frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d} + \frac{d}{d}\right)} - \frac{d}{d}}$

is 21 by counting.