A handsome number is a positive integer equal to a sum of powers of its decimal
digits, with a positive
integer exponent chosen independently for each digit. If the digits of 
are 
, ..., 
, then there exist positive
integers 
, ..., 
such that
 |
The first few handsome numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, ... (OEIS A007532). For example, 
and 
.
The name handsome number was proposed by Rivera in his Puzzle 15, which also considers representations allowing zero exponents. The positive integer exponent convention is used here. The name powerful number has also been used for handsome numbers, separately from its usual definition in terms of prime factors.
The positive narcissistic numbers are a special case in which each digit is raised to the number of digits. The inclusion is strict since 24 is a handsome number but is not a narcissistic number.
For counting representations, each digit is assigned the numerical value of its power. Changing an exponent
without changing that value does not create a new representation. Assignments to
unequal digits remain distinct even when the resulting
summands have the same values, as in 
.
If assignments that differ only by exchanging powers among identical digits are identified, the first few handsome
numbers with exactly two representations are 264, 373, 375, 935, 2139, 2203, 2223,
2243, 2245, 2263, 2336, 2352, 2356, 2372, 2376, 2394, 2407, 2427, ... (OEIS A050240).
With the same treatment of identical digits, requiring
at least two representations also includes 2332 and 2334, which have three and four
representations, respectively. The number 2132 has only one representation under
this convention, 
. If assignments
of different powers to identical digit
positions are instead counted separately, the first few handsome numbers with exactly
two representations are 224, 226, 264, 332, 334, 375, 377, 445, ... (OEIS A050241).
