Friedman number

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 = 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 = 8^6–3	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 × (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 = 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 = 4^5×2−4	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 = 2^E + 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 × 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 = 3⁴ × 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

$d=b-1$ be the greatest digit of the repdigit in base

$b$ . Then

${\begin{aligned}d\sum _{i=0}^{20}b^{i}&=\left(d+{\frac {d}{d}}\right)^{\left({\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}\right)\left({\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}+{\frac {d}{d}}\right)}-{\frac {d}{d}}\\&=(d+1)^{(5)(5)}-1\\&=b^{21}-1=(b-1)\sum _{i=0}^{20}b^{i}=d\sum _{i=0}^{20}b^{i}\end{aligned}}$

is a Friedman number. The number of digits in

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

is 21 by counting.

Friedman number

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