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

%I A373641 #27 Jul 05 2024 11:10:27
%S A373641 1,2,3,10,5,36,7,120,45,90,0,924,0,182,210,3860,0,3060,0,3800,420,0,0,
%T A373641 61824,300,0,3627,10584,0,25230,0,375968,0,0,1190,441000,0,0,0,426400,
%U A373641 0,70602,0,0,44550,0,0,11936496,1176,58800,0,0,0,1491102,0,1638560
%N A373641 The number of positive n-digit integers whose digit product is n.
%C A373641 Trivially, for the four single-digit primes p, a(p)=p.
%C A373641 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.
%H A373641 Graham Holmes, <a href="/A373641/b373641.txt">Table of n, a(n) for n = 1..10000</a>
%F A373641 a(n) = 0 <=> n in { A068191 }.
%F A373641 a(n) > 0 <=> n in { A002473 }.
%F A373641 a(n) = A163767(n) for n <= 9.
%e A373641 a(4) = 10: 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111.
%p A373641 b:= proc(n, t, i) option remember; `if`(n=1, 1/t!, `if`(i<2, 0,
%p A373641       add(b(n/i^j, t-j, i-1)/j!, j=0..padic[ordp](n, i))))
%p A373641     end:
%p A373641 a:= n-> n!*b(n$2, 9):
%p A373641 seq(a(n), n=1..56);  # _Alois P. Heinz_, Jun 12 2024
%Y A373641 Cf. A002033, A002473, A068191, A074206, A163767.
%K A373641 nonn,base
%O A373641 1,2
%A A373641 _Graham Holmes_, Jun 12 2024