A392010 The smallest nonnegative number k such that n! does not contain k as a substring.

0, 0, 0, 0, 0, 3, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 7, 5, 3, 3, 10, 5, 6, 7, 11, 2, 2, 7, 10, 4, 10, 10, 11, 10, 10, 8, 10, 10, 9, 10, 10, 12, 10, 10, 10, 10, 11, 10, 13, 10, 10, 12, 11, 12, 10, 16, 10, 10, 10, 13, 17, 11, 10, 13, 12, 11, 10, 10, 13, 10

Offset

0,6

Links

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Example

a(4) = 0 as 4! = 24 which does not contain '0' as a substring.
a(5) = 3 as 5! = 120 which does not contains '3' as a substring.
a(23) = 10 as 23! = 25852016738884976640000 which does not contain '10' as a substring.

Mathematica

a[n_]:=Module[{k=0}, While[SequenceCount[IntegerDigits[n!], IntegerDigits[k]]>0, k++]; k]; Array[a, 73, 0] (* James C. McMahon, Jan 28 2026 *)

Prog

(Python)
from gmpy2 import digits, fac
from itertools import count
def a(n):
    s = digits(fac(n))
    return next(k for k in count(0) if digits(k) not in s)
print([a(n) for n in range(73)]) # Michael S. Branicky, Jan 24 2026
(PARI) a(n) = my(k=0, sn=Str(n!)); while (#strsplit(sn, Str(k)) >= 2, k++); k; \\ Michel Marcus, Jan 24 2026

Crossrefs

Cf. A000142, A346120, A392011, A137578.

Sequence in context: A375932 A066636 A137578 * A200146 A319095 A109390

Adjacent sequences: A392007 A392008 A392009 * A392011 A392012 A392013

Keyword

nonn,base

Author

Scott R. Shannon, Dec 26 2025

Status

approved