A382217 a(n) is the least k > 0 such that the factorial base expansion of n starts with that of k.

1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 1, 4, 4, 5, 5, 16, 17, 18, 19, 20, 21, 22, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 16, 16, 17, 17, 64, 65, 66, 67, 68, 69, 70, 71, 18, 18, 19, 19, 76, 77, 20

Offset: 1

Author

Rémy Sigrist, Mar 19 2025

Keywords

• nonn
• base

Examples

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(n)  fact(a(n))
  --  ----  -------  ----------
   1     1  1        1
   2     1  1,0      1
   3     1  1,1      1
   4     4  2,0      2,0
   5     5  2,1      2,1
   6     1  1,0,0    1
   7     1  1,0,1    1
   8     1  1,1,0    1
   9     1  1,1,1    1
  10     1  1,2,0    1
  11     1  1,2,1    1
  12     4  2,0,0    2,0
  13     4  2,0,1    2,0
  14     5  2,1,0    2,1
  15     5  2,1,1    2,1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000030, A265334, A382184, A382218 (fixed points).

Programs

PARI

\\ See Links section.

Formula

a(n) = 1 iff n belongs to A265334.

a(n) <= A382184(n).

a(n) = n iff n belongs to A382218.