A351987 -id:A351987 - cp's OEIS frontend

A351988 In the factorial base expansion of n, arrange digits in decreasing order.

0, 1, 2, 3, 4, 5, 6, 8, 8, 9, 14, 15, 12, 14, 14, 15, 16, 17, 18, 20, 20, 21, 22, 23, 24, 30, 30, 32, 54, 56, 30, 32, 32, 33, 56, 57, 54, 56, 56, 57, 62, 63, 78, 80, 80, 81, 86, 87, 48, 54, 54, 56, 60, 62, 54, 56, 56, 57, 62, 63, 60, 62, 62, 63, 64, 65, 84, 86

Offset: 0

Rémy Sigrist, Feb 27 2022

This sequence is to factorial base what A004186 is to decimal base.

			For n = 1664:
- the factorial base expansion of 1664 is "214110",
- arranging these digits in decreasing order gives "421110",
- so a(1664) = 4*6! + 2*5! + 1*4! + 1*3! + 1*2! + 0*1! = 3152.

Cf. A004186 (decimal analog), A073138 (binary analog), A108731, A319651 (ternary analog), A351987.

max = 5; b = MixedRadix[Range[max, 2, -1]]; a[n_] := FromDigits[Sort[IntegerDigits[n, b], Greater], b]; Array[a, max!, 0] (* Amiram Eldar, Feb 28 2022 *)

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

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

a(n) >= n with equality iff n belongs to A351987.

A381770 a(n) is the least k > 0 such that the factorial base expansion of k*n has digits in nonincreasing order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 3, 2, 2, 3, 1, 2, 1, 1, 3, 3, 2, 3, 3, 2, 2, 14, 2, 2, 2, 2, 2, 2, 1, 6, 3, 2, 2, 8, 1, 2, 1, 1, 2, 2, 1, 5, 1, 1, 1, 1, 4, 8, 4, 6, 6, 8, 1, 6, 4, 2, 2, 9, 1, 8, 1, 1, 7, 8, 1, 5, 1

Offset: 0

Rémy Sigrist, Mar 07 2025

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

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

Cf. A223475, A351987, A381771.

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););); }

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 for k in count(1) if is_non_inc(k*n))
print([a(n) for n in range(87)]) # Michael S. Branicky, Mar 09 2025

a(n) <= (n-1)! for any n > 0.

a(n) = 1 iff n belongs to A351987.

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.

Original entry on oeis.org

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

Rémy Sigrist, Mar 07 2025

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

			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

Cf. A223474, A351987, A381770.

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););); }

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

a(n) = A381770(n) * n.
a(n) <= n!.
a(n) = n iff n belongs to A351987.

cp's OEIS Frontend

A351988 In the factorial base expansion of n, arrange digits in decreasing order.

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A381770 a(n) is the least k > 0 such that the factorial base expansion of k*n has digits in nonincreasing order.

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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.

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Programs

PARI

Python

Formula