A341880 Number of ordered factorizations of n into 4 factors > 1.

1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 12, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 16, 1, 0, 0, 12, 0, 0, 0, 4, 0, 12, 0, 0, 0, 0, 0, 40, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 0, 16

Offset: 16

Author

Ilya Gutkovskiy, Feb 22 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 16..20000
• Eric Weisstein's World of Mathematics, Ordered Factorization

Crossrefs

Column k=4 of A251683.

Cf. A000005, A007425, A007426, A070824, A074206, A200221, A341881, A341882.

Programs

Maple

b:= proc(n) option remember; series(x*(1+add(b(n/d),
      d=numtheory[divisors](n) minus {1, n})), x, 5)
    end:
a:= n-> coeff(b(n), x, 4):
seq(a(n), n=16..112);  # Alois P. Heinz, Feb 22 2021

Mathematica

b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
     {d, Divisors[n] ~Complement~ {1, n}}]), {x, 0, 5}];
a[n_] := Coefficient[b[n], x, 4];
Table[a[n], {n, 16, 112}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

Formula

Dirichlet g.f.: (zeta(s) - 1)^4.

a(n) = 6 * A000005(n) - 4 * A007425(n) + A007426(n) - 4 for n > 1.