A295923 - OEIS

A295923

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

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 10, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 13, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 29, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 10, 2, 1, 11

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OFFSET

1,4

COMMENTS

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) factorizations of d.

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

The a(16) = 10 twice-factorizations are (2*2*2*2), (2*2*4), (2*8), (4*4), (16), (2*2)*(2*2), (2*2)*(4), (4)*(2*2), (4)*(4), (2)*(2)*(2)*(2).

MATHEMATICA

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

a[n_]:=Sum[Length[postfacs[n^(1/g)]]^g, {g, Divisors[GCD@@FactorInteger[n][[All, 2]]]}];

Array[a, 50]

CROSSREFS

Cf. A000005, A001055, A052409, A052410, A279787, A281113, A284639, A295920, A295924, A295931, A295935.

Sequence in context: A046645 A284639 A320887 * A325806 A016470 A369068

Adjacent sequences: A295920 A295921 A295922 * A295924 A295925 A295926

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 30 2017

STATUS

approved