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A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.

%I A384445 #5 Jun 10 2025 19:43:03
%S A384445 5,6,7,10,23,43,74,125,199,305,449,637,885,1216,1649,2184,2852,3664,
%T A384445 4657,5863,7298,9002,10993,13312,16000,19084,22613,26606,31120,36192,
%U A384445 41867,48220,55317,63232,72022,81746,92479,104282,117229,131393,146843,163652,181892
%N A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.
%H A384445 Felix Huber, <a href="/A384445/b384445.txt">Table of n, a(n) for n = 1..200</a>
%e A384445 a(3) = 7 because exactly for the 7 multisets with 3 digits {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 0, 5}, {0, 0, 7}, {0, 2, 2} and {1, 2, 3} their sum equals the product of the prime digits.
%e A384445 a(4) = 10 because exactly for the 10 multisets with 4 digits {0, 0, 0, 1}, {0, 1, 2, 3}, {1, 2, 4, 7}, {1, 3, 5, 6}, {0, 0, 0, 2}, {0, 0, 2, 2}, {0, 0, 0, 3}, {0, 0, 0, 5}, {5, 5, 6, 9} and {0, 0, 0, 7} their sum equals the product of the prime digits.
%p A384445 f:=proc(p,n)
%p A384445     local c,d,i,l,m,r,s,t,u,w,x,y,z;
%p A384445     m:={0,1,4,6,8,9};
%p A384445     t:=seq(cat(x,i),i in m);
%p A384445     y:={l='Union'(t),w='Set'(l),t=~'Atom'};
%p A384445     d:=(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))};
%p A384445     Order:=p+1;
%p A384445     r:=combstruct:-agfseries(y,d,'unlabeled',z,[[u,s]])[w(z,u)];
%p A384445     r:=collect(convert(r,'polynom'),[z,u],'recursive');
%p A384445     c:=coeff(r,z,p);
%p A384445     coeff(c,u,n)
%p A384445 end proc:
%p A384445 A384445:=proc(n)
%p A384445     local a,k,m,s,p,j,L;
%p A384445     a:=1;
%p A384445         for k from 9*n to 1 by -1 do
%p A384445             L:=ifactors(k)[2];
%p A384445             m:=nops(L);
%p A384445             if m>0 and L[m,1]<=7 then
%p A384445                 p:=n-add(L[j,2],j=1..m);
%p A384445                 s:=k-add(L[j,1]*L[j,2],j=1..m);
%p A384445                 if s=0 and p>=0 then
%p A384445                     a:=a+1
%p A384445                 elif p>0 and s>0 then
%p A384445                     a:=a+f(p,s)
%p A384445                 fi
%p A384445             fi
%p A384445 	od;
%p A384445 	return a
%p A384445 end proc;
%p A384445 seq(A384445(n),n=1..43);
%Y A384445 Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444, A384505.
%K A384445 nonn,base
%O A384445 1,1
%A A384445 _Felix Huber_, Jun 03 2025