A382177 a(n) is the least k > 1 such that the factorial base expansion of k*n starts with that of n while the remaining digits are zeros.

2, 2, 3, 10, 3, 312, 4, 18, 18, 96, 96, 600, 4, 6168960, 6120, 18, 18, 11017036800, 4, 56229997824000, 114, 760, 68947200, 18, 5, 14544, 141120, 192, 13320, 9092075324665919034015350784000000, 28, 520412336961032355840000, 27, 1400, 199584000, 116496, 180

Offset: 0

Author

Rémy Sigrist, Mar 17 2025

Keywords

• nonn
• base

Comments

This sequence is well defined: for any n > 0 and m >= 0, A153880^m(n) (where A153880^m denotes the m-th iterate of A153880) is a multiple of (m+1)! whose factorial base expansion starts with that of n while the remaining digits are zeros, so for m sufficiently large, n will divide (m+1)! and hence this value.

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       10  1,1      1,1,0,0
   4        3  2,0      2,0,0
   5      312  2,1      2,1,0,0,0,0
   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       96  1,1,1    1,1,1,0,0,0
  10       96  1,2,0    1,2,0,0,0,0
  11      600  1,2,1    1,2,1,0,0,0,0
  12        4  2,0,0    2,0,0,0
  13  6168960  2,0,1    2,0,1,0,0,0,0,0,0,0,0
  14     6120  2,1,0    2,1,0,0,0,0,0,0
  15       18  2,1,1    2,1,1,0,0
		

Links

• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A153880, A382178.

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) = { my (m = n); while (1, m = A153880(m); if (m==0, return (2), m%n==0, return (m/n));); }

Formula

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