OFFSET
2,1
COMMENTS
1 and 2 are factorions of every integer number base, since 1 = 1! and 2 = 2!. Thus every integer number base has at least 2 factorions. - Michael De Vlieger, Nov 23 2015
A factorion is an integer which is equal to the sum of factorials of its digits. See A193163 for the list of all factorions in base n. - M. F. Hasler, Nov 25 2015
LINKS
Eric Weisstein's World of Mathematics, Factorion
EXAMPLE
a(6) = 4 because base 6 has the factorions {1, 2, 25, 26}. Expressed in base 6 these are {1, 2, 41, 42}. 1! = 1 and 2! = 2 and are factorions in every integer base b >= 2. Additionally, 4! + 1! = 24 + 1 = 25 and 4! + 2! = 24 + 2 = 26. - Michael De Vlieger, Nov 23 2015
a(2) = 2 = #{ 1, 2 }, indeed 1 = 1! and 2 = 10[2] = 1! + 0! and there cannot be any other since the sum of factorials of the binary digits equals the number of these digits, and from 3 on all numbers are larger than the number of their binary digits. - M. F. Hasler, Nov 25 2015
MATHEMATICA
Table[Length@ Select[Range[n Factorial[n - 1]], Total@ Map[Factorial, #] &@ IntegerDigits[#, n] == # &], {n, 2, 10}] (* Michael De Vlieger, Nov 23 2015 *)
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Eric M. Schmidt (based on data from A193163), Nov 16 2015
STATUS
approved