A348138 For any number n with factorial base representation (d_1, ..., d_k), for i = 1..k, let f_i be the number of i's in (d_1, ..., d_k); the factorial base representation of a(n) is (f_1, ..., f_k).

0, 1, 2, 4, 1, 3, 6, 12, 12, 18, 8, 14, 2, 8, 8, 14, 4, 10, 1, 7, 7, 13, 3, 9, 24, 48, 48, 72, 30, 54, 48, 72, 72, 96, 54, 78, 30, 54, 54, 78, 36, 60, 26, 50, 50, 74, 32, 56, 6, 30, 30, 54, 12, 36, 30, 54, 54, 78, 36, 60, 12, 36, 36, 60, 18, 42, 8, 32, 32, 56

Offset: 0

Author

Rémy Sigrist, Oct 02 2021

Keywords

• nonn
• base

Comments

Leading zeros in factorial base representation of n are ignored.

Examples

			The first terms, in decimal and in factorial base, are:
  n   a(n)  f(n)  f(a(n))
  --  ----  ----  -------
   0     0     0        0
   1     1     1        1
   2     2    10       10
   3     4    11       20
   4     1    20        1
   5     3    21       11
   6     6   100      100
   7    12   101      200
   8    12   110      200
   9    18   111      300
  10     8   120      110
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..5040
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000142, A108731, A034968, A060130, A153880.

Programs

PARI

a(n) = { my (f=[]); for (r=2, oo, if (n==0, return (sum(k=1, #f, f[k]*(#f-k+1)!)), f=concat(f, 0); my (d=n%r); n\=r; if (d, f[d]++))) }

Formula

a(n) = n iff n = 0 or n is a factorial number (A000142).

A034968(a(n)) = A060130(n).

a(A153880(n)) = A153880(a(n)).