A381771 For any n > 0, a(n) is the least positive multiple of n whose factorial base expansion has digits in nonincreasing order; a(0) = 0.

0, 1, 2, 3, 4, 5, 6, 14, 8, 9, 20, 22, 12, 65, 14, 15, 16, 17, 18, 57, 20, 21, 22, 23, 24, 150, 78, 54, 56, 87, 30, 62, 32, 33, 102, 105, 72, 111, 114, 78, 80, 574, 84, 86, 88, 90, 92, 94, 48, 294, 150, 102, 104, 424, 54, 110, 56, 57, 116, 118, 60, 305, 62, 63

Offset

0,3

Comments

The sequence is well defined as for any n > 0, the factorial base expansion of n! has digits in nonincreasing order.

Links

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

Index entries for sequences related to factorial base representation

Formula

a(n) = A381770(n) * n.

a(n) <= n!.

a(n) = n iff n belongs to A351987.

Example

The first terms, alongside their factorial base expansion, are:
  n   a(n)  fact(a(n))
  --  ----  ----------
   0     0  0
   1     1  1
   2     2  1,0
   3     3  1,1
   4     4  2,0
   5     5  2,1
   6     6  1,0,0
   7    14  2,1,0
   8     8  1,1,0
   9     9  1,1,1
  10    20  3,1,0
  11    22  3,2,0
  12    12  2,0,0
  13    65  2,2,2,1
  14    14  2,1,0
  15    15  2,1,1

Prog

(PARI) is(n) = { my (p = -1); for (r = 2, oo, if (n==0, return (1), p > p = n%r, return (0)); n \= r; ); }
a(n) = { for (k = 1, oo, if (is(k*n), return (k*n); ); ); }
(Python)
from itertools import count
def facbase(n, i=2): return [n] if n < i else [*facbase(n//i, i=i+1), n%i]
def is_non_inc(n): return (fb:=facbase(n)) == sorted(fb, reverse=True)
def a(n): return next(k*n for k in count(1) if is_non_inc(k*n))
print([a(n) for n in range(64)]) # Michael S. Branicky, Mar 09 2025

Crossrefs

Cf. A223474, A351987, A381770.

Sequence in context: A065638 A364224 A204896 * A334733 A039016 A249102

Adjacent sequences: A381768 A381769 A381770 * A381772 A381773 A381774

Keyword

nonn,base

Author

Rémy Sigrist, Mar 07 2025

Status

approved