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Number of odd-length factorizations of n into factors > 1.
78

%I #12 Jan 12 2021 19:48:15

%S 0,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,1,2,2,1,2,1,4,1,1,

%T 1,4,1,1,1,3,1,2,1,2,2,1,1,6,1,2,1,2,1,3,1,3,1,1,1,5,1,1,2,5,1,2,1,2,

%U 1,2,1,8,1,1,2,2,1,2,1,6,2,1,1,5,1,1,1

%N Number of odd-length factorizations of n into factors > 1.

%H Alois P. Heinz, <a href="/A339890/b339890.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) + A339846(n) = A001055(n).

%e The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:

%e 24 48 60 72 96 120

%e 2*2*6 2*3*8 2*5*6 2*4*9 2*6*8 3*5*8

%e 2*3*4 2*4*6 3*4*5 2*6*6 3*4*8 4*5*6

%e 3*4*4 2*2*15 3*3*8 4*4*6 2*2*30

%e 2*2*12 2*3*10 3*4*6 2*2*24 2*3*20

%e 2*2*2*2*3 2*2*18 2*3*16 2*4*15

%e 2*3*12 2*4*12 2*5*12

%e 2*2*2*3*3 2*2*2*2*6 2*6*10

%e 2*2*2*3*4 3*4*10

%e 2*2*2*3*5

%p g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+

%p `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),

%p d=numtheory[divisors](n) minus {1, n}))

%p end:

%p a:= n-> `if`(n<2, 0, g(n$2, 1)):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Dec 30 2020

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

%t Table[Length[Select[facs[n],OddQ@Length[#]&]],{n,100}]

%Y The case of set partitions (or n squarefree) is A024429.

%Y The case of partitions (or prime powers) is A027193.

%Y The ordered version is A174726 (even: A174725).

%Y The remaining (even-length) factorizations are counted by A339846.

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A001055 counts factorizations, with strict case A045778.

%Y A027193 counts partitions of odd length, ranked by A026424.

%Y A058695 counts partitions of odd numbers, ranked by A300063.

%Y A160786 counts odd-length partitions of odd numbers, ranked by A300272.

%Y A316439 counts factorizations by product and length.

%Y A340101 counts factorizations into odd factors.

%Y A340102 counts odd-length factorizations into odd factors.

%Y Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.

%K nonn

%O 1,8

%A _Gus Wiseman_, Dec 28 2020