A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷) and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 73 + 4. The first few base 10 Friedman numbers are:
- 25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3159 (sequence A036057 in OEIS).
Parentheses can be used in the expressions, but only to override the default operator precedence, for example, in 1024 = (4 − 2)10. 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.
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, we can arrange 127 = 27 − 1 as 127 = −1 + 27. The first nice Friedman numbers are:
- 127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 (sequence A080035 in OEIS).
Currently, 81 zeroless pandigital Friedman numbers are known. Two of them are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34, both discovered by Mike Reid and Philippe Fondanaiche. Only one of the 81 known zeroless pandigital Friedman numbers is nice: 268435179 = −268 + 4(3×5 − 17) − 9.
From the observation that all numbers of the form 25×102n can be written as 500...02 with n 0's, we can find strings of consecutive Friedman numbers. Friedman gives the example of 250068 = 5002 + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099.
It seems that all powers of 5 are Friedman numbers.
Fondanaiche thinks the smallest repdigit nice Friedman number is 99999999 = (9 + 9/9)9−9/9 − 9/9. Brandon Owens proved that repdigits of more than 24 digits are nice Friedman numbers in any base.
Vampire numbers are a type of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60.
Read more about Friedman Number: Finding 2-digit Friedman Numbers, Friedman Numbers Using Roman Numerals
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