A327695 - OEIS

A327695

Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 2, 0, 4, 0, 2, 2, 1, 0, 3, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 6, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 4, 0, 1, 2, 2, 1, 4, 0, 3, 0, 1, 0, 6, 1, 1, 1

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OFFSET

1,12

LINKS

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

FORMULA

a(n) = A327399(n) - A089723(n).

EXAMPLE

The factorizations of 6, 12, 30, 48, 60, 180, and 210:

(2*3) (3*4) (5*6) (3*16) (3*20) (4*45) (3*70)

(2*2*3) (2*15) (3*4*4) (4*15) (5*36) (5*42)

(3*10) (2*2*2*2*3) (5*12) (9*20) (6*35)

(2*3*5) (3*4*5) (4*5*9) (7*30)

(2*2*15) (5*6*6) (10*21)

(2*2*3*5) (2*2*45) (14*15)

(3*3*20) (2*105)

(2*2*5*9) (5*6*7)

(3*3*4*5) (2*3*35)

(2*2*3*3*5) (2*5*21)

(2*7*15)

(3*5*14)

(3*7*10)

(2*3*5*7)

MATHEMATICA

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

Table[Length[Select[facs[n], CoprimeQ@@Union[#]&]], {n, 100}]

CROSSREFS

Factorizations that are constant or whose distinct parts are pairwise coprime are counted by A327399.

Numbers with pairwise coprime distinct prime indices are A304711.

Cf. A001055, A089723, A281116, A318721, A302569, A319269, A327407, A327517.

Sequence in context: A332036 A242444 A236441 * A345446 A354911 A367098

Adjacent sequences: A327692 A327693 A327694 * A327696 A327697 A327698

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 22 2019

STATUS

approved