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Number of twice-factorizations of n of where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

allocated for Gus WisemanNumber of twice-factorizations of n of type (P,Q,R).

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 9, 2, 2, 2, 9, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 13, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 11, 1, 2, 4, 16, 2, 5, 1, 4, 2, 5, 1, 18, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 11, 2

1,4

a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.

Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).

The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).

sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];

Table[Sum[Product[DivisorSigma[0, GCD@@FactorInteger[d][[All, 2]]], {d, fac}], {fac, sfs[n]}], {n, 100}]

Cf. A001055, A005117, A045778, A063834, A089723, A279786, A281113, A296120, A296118.

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nonn

Gus Wiseman, Dec 05 2017

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allocated for Gus Wiseman

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