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A360972 Number of n-digit zeroless numbers whose digit sum is larger than the product of all digits.

%I A360972 #13 Feb 27 2023 15:25:54
%S A360972 0,0,17,28,51,91,189,302,513,985,1461,1904,3155,4616,7211,10236,12709,
%T A360972 15573,21463,31161,56306,70904,85570,100626,135203,155951,188059,
%U A360972 293887,434106,522755,612837,888740,1013337,1182809,1559292,1755888,1971286,2208087,2491034
%N A360972 Number of n-digit zeroless numbers whose digit sum is larger than the product of all digits.
%H A360972 Alois P. Heinz, <a href="/A360972/b360972.txt">Table of n, a(n) for n = 0..1000</a>
%F A360972 a(n) = |{ k in { A052382 } : A055642(k) = n and A007953(k) > A007954(k) }|.
%e A360972 a(2) = 17: 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91.
%e A360972 a(3) = 28: 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 141, 151, 161, 171, 181, 191, 211, 212, 221, 311, 411, 511, 611, 711, 811, 911.
%e A360972 a(4) = 51: 1111, 1112, 1113, 1114, ..., 6111, 7111, 8111, 9111.
%p A360972 b:= proc(n, i, s, p) option remember;
%p A360972       `if`(s+n*i<=p, 0, `if`(n=0 or i=1, 1/n!,
%p A360972        add(b(n-j, i-1, s+i*j, p*i^j)/j!, j=0..n)))
%p A360972     end:
%p A360972 a:= n-> b(n, 9, 0, 1)*n!:
%p A360972 seq(a(n), n=0..40);
%Y A360972 Cf. A007953, A007954, A052382, A055642, A360538.
%K A360972 nonn,base
%O A360972 0,3
%A A360972 _Alois P. Heinz_, Feb 27 2023