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

0,1

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.

Links

Table of n, a(n) for n=0..36.

Index entries for sequences related to factorial base representation

Formula

a(k!) = 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)*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

Prog

(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)); ); }

Crossrefs

Cf. A153880, A382178.

Sequence in context: A091044 A079661 A220644 * A338372 A376723 A153920

Adjacent sequences: A382174 A382175 A382176 * A382178 A382179 A382180

Keyword

nonn,base

Author

Rémy Sigrist, Mar 17 2025

Status

approved