A385673 The smallest number k such that k! contains the string concatenation of 0 to n as a substring.

5, 12, 32, 106, 363, 992, 992, 4255, 20527, 103638, 274650

Offset

0,1

Links

Table of n, a(n) for n=0..10.

Example

a(0) = 5 as 5! = 0 which contains '0' as a substring.
a(1) = 12 as 12! = 479001600 which contains '01' as a substring.
a(2) = 32 as 32! = 263130836933693530167218012160000000 which contains '012' as a substring.

Mathematica

a[n_]:=Module[{k=1}, While[SequenceCount[IntegerDigits[k!], Range[0, n]]<1, k++]; k]; Array[a, 8, 0] (* James C. McMahon, Jan 27 2026 *)

Prog

(Python)
from gmpy2 import mpz, digits
from itertools import count
def A385673(n):
    f = mpz(1)
    t = ''.join(digits(i) for i in range(n+1))
    l = len(t)
    for k in count(1):
        f *= k
        s = digits(f)
        for i in range(1, len(s)-l+1):
            if s[i:i+l] == t:
                return k # Chai Wah Wu, Jan 26 2026

Crossrefs

Cf. A000142, A392010, A392011, A346120, A137578.

Sequence in context: A192303 A301785 A066280 * A333886 A192243 A292104

Adjacent sequences: A385670 A385671 A385672 * A385674 A385675 A385676

Keyword

nonn,base,hard,more

Author

Scott R. Shannon, Jan 25 2026

Extensions

a(10) from Chai Wah Wu, Jan 26 2026

Status

approved