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A296119 Number of ways to choose a strict factorization of each factor in a factorization of n. 8
1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 21, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 23, 3, 12, 1, 7, 3, 12, 1, 50, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

FORMULA

Dirichlet g.f.: 1/Product_{n > 1}(1 - A045778(n)/n^s).

EXAMPLE

The a(24) = 16 twice-factorizations:

(2)*(2)*(2)*(3),

(2)*(2)*(2*3), (2)*(2)*(6), (2)*(3)*(4),

(2)*(2*6), (2)*(3*4), (2)*(12), (3)*(2*4), (3)*(8), (4)*(2*3), (4)*(6),

(2*3*4), (2*12), (3*8), (4*6), (24).

MATHEMATICA

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

Table[Sum[Times@@(Length[Select[facs[#], UnsameQ@@#&]]&/@fac), {fac, facs[n]}], {n, 100}]

PROG

(PARI)

A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));

A296119(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A045778(d)*A296119(n/d, d)))); \\ Antti Karttunen, Oct 08 2018

CROSSREFS

Cf. A000009, A005117, A045778, A270995, A281113, A294788, A296118, A296120, A296121.

Sequence in context: A079616 A292587 A097283 * A300836 A118314 A002033

Adjacent sequences:  A296116 A296117 A296118 * A296120 A296121 A296122

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 05 2017

STATUS

approved

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Last modified February 26 11:49 EST 2020. Contains 332279 sequences. (Running on oeis4.)