Friedman number
A Friedman number is an integer which in a given base is the result of an expression using 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
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 lame Friedman numbers such as 24 = (24). Leading zeroes can not be used, since that would also result in lame Friedman numbers, such as 001729 = 1700 + 29.
Currently, two zeroless pandigital Friedman numbers are known: 123456789 = ((86 + 2 * 7)5 - 91) / 34, and 987654321 = (8 * (97 + 6/2)5 + 1) / 34, both discovered by Mike Reid and Philippe Fondanaiche.
From the observation that all powers of 5 appear to be Friedman numbers, 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 250010 to 250099.
A nice 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. All the expressions for nice Friedman numbers less than 10000 involve addition and substraction. The first few nice Frieman 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
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.
Algorithms for finding Friedman numbers
There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as mb + n, where b is the base and m, n are integers between -1 and b, we need only check each possible combination of m and n against the equalities mb + n == mn, mb + n == mn, and mb + n == nm to see which ones return true. We need not concern ourselves with m + n, since a little reflection will show that mb + n == m +n always returns false. From there it becomes obvious we need not concern ourselves with expressions like m - n and m/n.
When we get to 3-digit numbers, the concept remains the same, only that there are more possible expressions to check. Representing a three digit number as kb2 + mb + n, there are more expressions to check, for starters, km + n, kn + m, km + n, n * (kb + m), etc.
Friedman numbers using Roman numerals
In a trivial sense, all Roman numerals with more than one symbol are Friedman numbers. The expression is created by simply inserting + signs into the numeral, and occassionally the - sign with slight rearrangement of the order of the symbols.
But Erich Friedman and Robert Happleberg have done some research into Roman numeral Friedman numbers for which the expression uses some of the other operators. Their first discovery was the nice Friedman number 8, since VIII = (V - I) * II. They have also found many Roman numeral Friedman numbers for which the expression uses exponentiation, e.g., 256 since CCLVI = IVCC/L.
The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case with positional notation numbering systems) but with the numbers of symbols it has. So, for example, it is much tougher to figure out whether 137 (CXLVII) is a Friedman number in Roman numerals than it is to make the same determination for 1001 (MI). With Roman numerals, one can at least derive quite a few Friedman expressions from any new expression one discovers. Friedman and Happleberg have shown that any number ending in VIII is a Friedman number based on the expression given above, for instance.