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

0,4

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Index entries for sequences related to factorial base representation

Formula

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

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

Example

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

Prog

(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); }

Crossrefs

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

Sequence in context: A051993 A377739 A174530 * A280490 A350091 A135347

Adjacent sequences: A382181 A382182 A382183 * A382185 A382186 A382187

Keyword

nonn,base

Author

Rémy Sigrist, Mar 17 2025

Status

approved