A260651 Number of factorions in base n.

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, 3, 3, 3, 2

Offset: 2

Author

Eric M. Schmidt (based on data from A193163), Nov 16 2015

Keywords

• nonn
• base
• more

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

Examples

			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
		

Links

• Eric Weisstein's World of Mathematics, Factorion

Crossrefs

Cf. A014080, A193163.

Programs

Mathematica

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