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Number of even-length factorizations of n into factors > 1.
77

%I #18 Oct 22 2023 15:15:09

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

%T 1,5,0,1,1,4,0,3,0,2,2,1,0,6,1,2,1,2,0,4,1,4,1,1,0,6,0,1,2,6,1,3,0,2,

%U 1,3,0,8,0,1,2,2,1,3,0,6,3,1,0,6,1,1,1,4,0,6,1,2,1,1,1,10,0,2,2,5,0,3,0,4,3

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

%H Antti Karttunen, <a href="/A339846/b339846.txt">Table of n, a(n) for n = 1..65537</a>

%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) + A339890(n) = A001055(n).

%e The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:

%e 2*6 2*8 3*8 4*9 6*8 8*9 2*48 2*60

%e 3*4 4*4 4*6 6*6 2*24 2*36 3*32 3*40

%e 2*2*2*2 2*12 2*18 3*16 3*24 4*24 4*30

%e 2*2*2*3 3*12 4*12 4*18 6*16 5*24

%e 2*2*3*3 2*2*2*6 6*12 8*12 6*20

%e 2*2*3*4 2*2*2*9 2*2*3*8 8*15

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

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

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

%e 2*2*2*2*2*3 2*2*2*15

%e 2*2*3*10

%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=1, 1, g(n$2, 0)):

%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],EvenQ@Length[#]&]],{n,100}]

%o (PARI) A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ _Antti Karttunen_, Oct 22 2023

%Y The case of set partitions (or n squarefree) is A024430.

%Y The case of partitions (or prime powers) is A027187.

%Y The ordered version is A174725, odd: A174726.

%Y The odd-length factorizations are counted by A339890.

%Y A001055 counts factorizations, with strict case A045778.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A027187 counts partitions of even length, ranked by A028260.

%Y A058696 counts partitions of even numbers, ranked by A300061.

%Y A316439 counts factorizations by product and length.

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

%Y Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

%K nonn

%O 1,12

%A _Gus Wiseman_, Dec 28 2020

%E Data section extended up to a(105) by _Antti Karttunen_, Oct 22 2023