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

2,1

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

Table of n, a(n) for n=2..39.

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

Cf. A014080, A193163.

Sequence in context: A322997 A367315 A085561 * A116370 A378160 A261018

Adjacent sequences: A260648 A260649 A260650 * A260652 A260653 A260654

Keyword

nonn,base,more

Author

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

Status

approved