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

1,1

Links

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

Example

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.

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);

Crossrefs

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

Sequence in context: A308193 A179274 A104122 * A336857 A049467 A131503

Adjacent sequences: A384442 A384443 A384444 * A384446 A384447 A384448

Keyword

nonn,base

Author

Felix Huber, Jun 03 2025

Status

approved