A382178 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n.

2, 2, 3, 3, 3, 3, 4, 18, 18, 17, 17, 16, 4, 19, 19, 18, 18, 101, 4, 115, 114, 110, 110, 18, 5, 203, 199, 192, 189, 183, 28, 187, 27, 179, 177, 1341, 180, 176, 26, 170, 168, 165, 1320, 168, 1277, 1251, 162, 159, 5, 1649, 204, 1598, 1579, 1551, 200, 197, 195

Offset: 0

Author

Rémy Sigrist, Mar 17 2025

Keywords

• nonn
• base

Comments

This sequence is well defined (see A382177).

Examples

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(n)  fact(a(n)*n)
  --  ----  -------  ------------
   0     2  0        0
   1     2  1        1,0
   2     3  1,0      1,0,0
   3     3  1,1      1,1,1
   4     3  2,0      2,0,0
   5     3  2,1      2,1,1
   6     4  1,0,0    1,0,0,0
   7    18  1,0,1    1,0,1,0,0
   8    18  1,1,0    1,1,0,0,0
   9    17  1,1,1    1,1,1,1,1
  10    17  1,2,0    1,2,0,1,0
  11    16  1,2,1    1,2,1,1,0
  12     4  2,0,0    2,0,0,0
  13    19  2,0,1    2,0,1,0,1
  14    19  2,1,0    2,1,0,1,0
  15    18  2,1,1    2,1,1,0,0
		

Links

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

Crossrefs

Cf. A153880, A382177.

Programs

PARI

A153880(n) = { my (v = 0, f = 1); for (r = 2, oo, if (n==0, return (v);); v += (n%r) * f *= r; n \= r;); }
a(n) = { if (n==0, return (2)); my (m = n, f = 1, e); for (r = 2, oo, m = A153880(m); f *= r; e = (-m) % n; if (e < f, return ((m+e)/n););); }

Formula

a(n) <= A382177(n).