Factorion
From Wikipedia, the free encyclopedia
A factorion is a natural number that equals the sum of the factorials of its decimal digits. For example, 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145.
There are just four factorions and they are 1, 2, 145 and 40585 (sequence A014080 in OEIS).
[edit] Upper bound
If n is a natural number of d digits that is a factorion, then 10d − 1 ≤ n ≤ 9!d and this fails to hold for d ≥ 8. Thus n has 7 digits at most and the first upper bound is 9,999,999. But the maximum sum of factorials of digits for a 7 digit number is 9!7 = 2540160, which is the second upper bound.
[edit] Code
Some Haskell code to compute factorions. Be sure to compile it with -O2 or your memory will be exhausted.
main = print $ filter (isFactorion 10) [1..2540160] factList = scanl (*) 1 [1..] quickFact n = factList !! n digitlist 0 _ = [] digitlist num base = let digit = mod num base in digit : digitlist (div (num - digit) base) base sumOfFactOfDigits base n = sum $ map quickFact $ digitlist n base isFactorion base n = n == sumOfFactOfDigits base n
