A296132 Number of twice-factorizations of n where the first factorization is constant and the latter factorizations are strict, i.e., type (P,R,Q).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 9, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 10, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

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) strict factorizations of d.

Links

Table of n, a(n) for n=1..87.

Example

The a(36) = 9 twice-factorizations are (2*3)*(2*3), (2*3)*(6), (6)*(2*3), (6)*(6), (2*3*6), (2*18), (3*12), (4*9), (36).

Mathematica

sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Length[sfs[n^(1/g)]]^g, {g, Divisors[GCD@@FactorInteger[n][[All, 2]]]}], {n, 100}]

Crossrefs

Cf. A000005, A001055, A005117, A045778, A052409, A052410, A089723, A279788, A281113, A284639, A295920, A295931.

Sequence in context: A359909 A319269 A320888 * A253557 A326191 A326189

Adjacent sequences: A296129 A296130 A296131 * A296133 A296134 A296135

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Status

approved