A392011 The smallest number k such that k! contains all numbers 0, 1, ..., 10^n-1 as substrings.

5, 23, 220, 2039, 20334, 202498, 2051859

Offset

0,1

Comments

Equivalently, contains all n-digit numbers as substrings (for n > 0). - Michael S. Branicky, Jan 27 2026

Links

Table of n, a(n) for n=0..6.

Formula

From Michael S. Branicky, Jan 24 2026: (Start)

a(n+1) >= a(n) for n >= 1, by construction.

a(n) = min_{k | A392010(k) >= 10^n}. (End)

Example

a(0) = 5 as 5! = 120 which contains 0 as a substring.
a(1) = 23 as 23! = 25852016738884976640000 which contains all numbers 0 to 9 as substrings.
a(2) = 220 as 220! = 228386...0000 (422 digits) which contains all numbers 0 to 99 as substrings.

Prog

(Python)
from gmpy2 import digits, mpz
from itertools import count
def a(n):
    T = [digits(i) for i in range(10**n)][::-1] # target strings
    fk = mpz(1)
    for k in count(0):
        dk = digits(fk)
        if all(t in dk for t in T):
            return k
        fk *= mpz(k+1)
print([a(n) for n in range(4)]) # Michael S. Branicky, Jan 24 2026, edited Jan 27 2026

Crossrefs

Cf. A000142, A392010, A346120, A137578.

Sequence in context: A173041 A156187 A322238 * A295080 A323993 A156555

Adjacent sequences: A392008 A392009 A392010 * A392012 A392013 A392014

Keyword

nonn,base,hard,more

Author

Scott R. Shannon, Dec 26 2025

Extensions

a(6) from Chai Wah Wu, Feb 14 2026

Status

approved