A260651 - OEIS

%I #17 Feb 16 2025 08:33:26

%S 2,2,3,3,4,2,2,3,4,5,2,3,3,4,3,5,2,2,2,3,2,3,4,2,4,4,3,2,3,2,4,2,6,3,

%T 3,3,3,2

%N Number of factorions in base n.

%C 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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorion.html">Factorion</a>

%e 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

%e 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

%t Table[Length@ Select[Range[n Factorial[n - 1]], Total@ Map[Factorial, #] &@ IntegerDigits[#, n] == # &], {n, 2, 10}] (* _Michael De Vlieger_, Nov 23 2015 *)

%Y Cf. A014080, A193163.

%K nonn,base,more,changed

%O 2,1

%A _Eric M. Schmidt_ (based on data from A193163), Nov 16 2015