A223474 -id:A223474 - cp's OEIS frontend

A223475 Least k such that the decimal representation of k*n has digits in nonincreasing order.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 2, 3, 3, 4, 1, 1, 1, 4, 3, 2, 2, 2, 3, 3, 1, 1, 1, 1, 13, 2, 2, 2, 2, 17, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 15, 13, 9, 9, 1, 1, 1, 1, 1, 1, 1, 13, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 84, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 86, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 5, 7, 5, 2, 5, 3, 4, 6, 1, 1, 75, 47, 38, 8, 45, 56, 8, 7, 5, 55, 5, 7

Offset: 1

Paul Tek, Mar 20 2013

			39*17 = 663 has digits in nonincreasing order, and no k < 17 has this property, hence a(39) = 17.

Giovanni Resta, Table of n, a(n) for n = 1..2000

a(n)*n yields sequence A223474.
Cf. A079339, A181061.
Cf. A009996.

a[n_] := a[nn_] := Block[{n = nn, f, w = Range@9, k = 1}, While[Mod[n, 10] == 0, n /= 10]; While[(f = Select[w, Max@ Differences@ IntegerDigits[n*#] <= 0 &, 1]) == {}, k++; w = Union@ Flatten@Table[ Select[d*10^(k-1) + w, Max@ Differences@ IntegerDigits[Mod[n*#, 10^k], 10, k] <= 0 &], {d, 0, 9}]]; f[[1]]]; Array[a, 123] (* faster than basic approach. Giovanni Resta, Mar 26 2013 *)

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.

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A223475 Least k such that the decimal representation 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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