A382184 a(n) is the least k >= 0 such that the factorial base expansion of n starts with that of k while the remaining digits are zeros.

0, 1, 1, 3, 4, 5, 1, 7, 3, 9, 10, 11, 4, 13, 5, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1, 25, 7, 27, 28, 29, 3, 31, 9, 33, 34, 35, 10, 37, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 4, 49, 13, 51, 52, 53, 5, 55, 15, 57, 58, 59, 16, 61, 17, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Author

Rémy Sigrist, Mar 17 2025

Keywords

• nonn
• base

Examples

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000265, A004151, A153880, A273670, A382177, A382217.

Programs

PARI

a(n) = { if (n, my (m = n, s = oo, d); for (r = 2, oo, if (m==0 || s==0, break, d = m%r, s = min(s, r-1-d);); m \= r;); if (s, my (v = 0); for (r = 2, oo, if (n==0, return (v), v += (n%r) * max(0, r-1-s)!; n \= r;);););); return (n); }

Formula

a(n) <= n with equality iff n = 0 or n belongs to A273670.

a(k!) = 1 for any k >= 0.