A341881 Number of ordered factorizations of n into 5 factors > 1.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 15

Offset

32,17

Links

Alois P. Heinz, Table of n, a(n) for n = 32..20000

Eric Weisstein's World of Mathematics, Ordered Factorization

Formula

Dirichlet g.f.: (zeta(s) - 1)^5.

a(n) = -10 * A000005(n) + 10 * A007425(n) - 5 * A007426(n) + A061200(n) + 5 for n > 1.

Maple

b:= proc(n) option remember; series(x*(1+add(b(n/d),
      d=numtheory[divisors](n) minus {1, n})), x, 6)
    end:
a:= n-> coeff(b(n), x, 5):
seq(a(n), n=32..128);  # 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, 6}];
a[n_] := Coefficient[b[n], x, 5];
Table[a[n], {n, 32, 128}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

Crossrefs

Column k=5 of A251683.

Cf. A000005, A007425, A007426, A061200, A070824, A074206, A200221, A341880, A341882.

Sequence in context: A241471 A152623 A020761 * A281462 A236239 A047752

Adjacent sequences: A341878 A341879 A341880 * A341882 A341883 A341884

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 22 2021

Status

approved