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%I A339890 #12 Jan 12 2021 19:48:15
%S A339890 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 A339890 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 A339890 1,2,1,8,1,1,2,2,1,2,1,6,2,1,1,5,1,1,1
%N A339890 Number of odd-length factorizations of n into factors > 1.
%H A339890 Alois P. Heinz, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EzMzk4OTAvYjMzOTg5MC50eHQ">Table of n, a(n) for n = 1..20000</a>
%F A339890 a(n) + A339846(n) = A001055(n).
%e A339890 The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
%e A339890   24      48          60       72          96          120
%e A339890   2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
%e A339890   2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
%e A339890           3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
%e A339890           2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
%e A339890           2*2*2*2*3            2*2*18      2*3*16      2*4*15
%e A339890                                2*3*12      2*4*12      2*5*12
%e A339890                                2*2*2*3*3   2*2*2*2*6   2*6*10
%e A339890                                            2*2*2*3*4   3*4*10
%e A339890                                                        2*2*2*3*5
%p A339890 g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
%p A339890       `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
%p A339890           d=numtheory[divisors](n) minus {1, n}))
%p A339890     end:
%p A339890 a:= n-> `if`(n<2, 0, g(n$2, 1)):
%p A339890 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 30 2020
%t A339890 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A339890 Table[Length[Select[facs[n],OddQ@Length[#]&]],{n,100}]
%Y A339890 The case of set partitions (or n squarefree) is A024429.
%Y A339890 The case of partitions (or prime powers) is A027193.
%Y A339890 The ordered version is A174726 (even: A174725).
%Y A339890 The remaining (even-length) factorizations are counted by A339846.
%Y A339890 A000009 counts partitions into odd parts, ranked by A066208.
%Y A339890 A001055 counts factorizations, with strict case A045778.
%Y A339890 A027193 counts partitions of odd length, ranked by A026424.
%Y A339890 A058695 counts partitions of odd numbers, ranked by A300063.
%Y A339890 A160786 counts odd-length partitions of odd numbers, ranked by A300272.
%Y A339890 A316439 counts factorizations by product and length.
%Y A339890 A340101 counts factorizations into odd factors.
%Y A339890 A340102 counts odd-length factorizations into odd factors.
%Y A339890 Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.
%K A339890 nonn
%O A339890 1,8
%A A339890 _Gus Wiseman_, Dec 28 2020

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