A384505 a(n) is the number of multisets of n positive decimal digits where the sum of the digits equals the product of the prime digits.

5, 1, 1, 3, 13, 20, 31, 51, 74, 106, 144, 188, 248, 331, 433, 535, 668, 812, 993, 1206, 1435, 1704, 1991, 2319, 2688, 3084, 3529, 3993, 4514, 5072, 5675, 6353, 7097, 7915, 8790, 9724, 10733, 11803, 12947, 14164, 15450, 16809, 18240, 19757, 21374, 23073, 24876, 26759

Offset: 1

Author

Felix Huber, Jun 11 2025

Keywords

• nonn
• base

Examples

			a(1) = 5 because exactly for the 5 multisets with 1 digits {1}, {2}, {3}, {5}, and {7} their sum equals the product of the prime digits.
a(2) = 1 because only for 1 multiset with 2 positive digits {2, 2} their sum equals the product of the prime digits: 2 + 2 = 2*2 = 4.
a(3) = 1 because only for 1 multiset with 3 positive digits {1, 2, 3} their sum equals the product of the prime digits: 1 + 2 + 3 = 2*3 = 6.
a(4) = 3 because exactly for the 3 multisets with 4 digits {1, 2, 4, 7}, {1, 3, 5, 6}, and {5, 5, 6, 9} their sum equals the product of the prime digits: 1 + 2 + 4 + 7 = 2 * 7 = 14, 1 + 3 + 5 + 6 = 3*5 = 15, 5 + 5 + 6 + 9 = 5*5 = 25.
		

Links

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

Crossrefs

Partial sums give A384445.

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

Programs

Maple

f:=proc(p,n)
    local i,l,m,s,t,u,w,x,z;
    m:={1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    Order:=p+1;
    coeff(coeff(collect(convert(combstruct:-agfseries({l='Union'(t),w='Set'(l),t=~'Atom'},(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))},'unlabeled',z,[[u,s]])[w(z,u)],'polynom'),[z,u],'recursive'),z,p),u,n)
end proc:
A384505:=proc(n)
    local a,k,m,s,p,j,L;
    if n=1 then
        5
    elif n=2 then
        1
    else
        a:=0;
        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 p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
	fi;
end proc;
seq(A384505(n),n=1..48);

Formula

a(n) = A384445(n) - A384445(n-1) for n > 1.