A373641 The number of positive n-digit integers whose digit product is n.

1, 2, 3, 10, 5, 36, 7, 120, 45, 90, 0, 924, 0, 182, 210, 3860, 0, 3060, 0, 3800, 420, 0, 0, 61824, 300, 0, 3627, 10584, 0, 25230, 0, 375968, 0, 0, 1190, 441000, 0, 0, 0, 426400, 0, 70602, 0, 0, 44550, 0, 0, 11936496, 1176, 58800, 0, 0, 0, 1491102, 0, 1638560

Offset: 1

Author

Graham Holmes, Jun 12 2024

Keywords

• nonn
• base

Comments

Trivially, for the four single-digit primes p, a(p)=p.

It's not possible by definition to have a digit product equal to a prime number greater than 10, so a(p)=0 for prime p > 10.

Examples

			a(4) = 10: 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111.
		

Links

• Graham Holmes, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002033, A002473, A068191, A074206, A163767.

Programs

Maple

b:= proc(n, t, i) option remember; `if`(n=1, 1/t!, `if`(i<2, 0,
      add(b(n/i^j, t-j, i-1)/j!, j=0..padic[ordp](n, i))))
    end:
a:= n-> n!*b(n$2, 9):
seq(a(n), n=1..56);  # Alois P. Heinz, Jun 12 2024

Formula

a(n) = 0 <=> n in { A068191 }.

a(n) > 0 <=> n in { A002473 }.

a(n) = A163767(n) for n <= 9.