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.

5, 6, 7, 10, 23, 43, 74, 125, 199, 305, 449, 637, 885, 1216, 1649, 2184, 2852, 3664, 4657, 5863, 7298, 9002, 10993, 13312, 16000, 19084, 22613, 26606, 31120, 36192, 41867, 48220, 55317, 63232, 72022, 81746, 92479, 104282, 117229, 131393, 146843, 163652, 181892

Offset: 1

Author

Felix Huber, Jun 03 2025

Keywords

• nonn
• base

Examples

			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.
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.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444, A384505.

Programs

Maple

f:=proc(p,n)
    local c,d,i,l,m,r,s,t,u,w,x,y,z;
    m:={0,1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    y:={l='Union'(t),w='Set'(l),t=~'Atom'};
    d:=(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))};
    Order:=p+1;
    r:=combstruct:-agfseries(y,d,'unlabeled',z,[[u,s]])[w(z,u)];
    r:=collect(convert(r,'polynom'),[z,u],'recursive');
    c:=coeff(r,z,p);
    coeff(c,u,n)
end proc:
A384445:=proc(n)
    local a,k,m,s,p,j,L;
    a:=1;
        for k from 9*n to 1 by -1 do
            L:=ifactors(k)[2];
            m:=nops(L);
            if m>0 and L[m,1]<=7 then
                p:=n-add(L[j,2],j=1..m);
                s:=k-add(L[j,1]*L[j,2],j=1..m);
                if s=0 and p>=0 then
                    a:=a+1
                elif p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
end proc;
seq(A384445(n),n=1..43);