A341882 -id:A341882 - cp's OEIS frontend

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

Ilya Gutkovskiy, Feb 22 2021

nonn

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

Column k=4 of A251683.

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

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

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 *)

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

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

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Feb 22 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 32..20000
Eric Weisstein's World of Mathematics, Ordered Factorization

Column k=5 of A251683.

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

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

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 *)

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

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

cp's OEIS Frontend

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

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