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

Offset: 1

Author

Gus Wiseman, Nov 30 2017

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

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

Programs

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]