%I #7 Dec 30 2020 19:58:03
%S 0,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,
%T 1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,1,2,1,1,1,2,1,1,1,1,3,
%U 1,1,1,1,1,2,1,1,2,1,1,1,1,1,2,1,1,2,1
%N Number of factorizations of 2n + 1 into an odd number of odd factors > 1.
%e The factorizations for 2n + 1 = 135, 225, 315, 405, 675, 1155, 1215:
%e 135 225 315 405 675 1155 1215
%e 3*5*9 5*5*9 5*7*9 5*9*9 3*3*75 3*5*77 3*5*81
%e 3*3*15 3*3*25 3*3*35 3*3*45 3*5*45 3*7*55 3*9*45
%e 3*5*15 3*5*21 3*5*27 3*9*25 5*7*33 5*9*27
%e 3*7*15 3*9*15 5*5*27 3*11*35 9*9*15
%e 3*3*3*3*5 5*9*15 5*11*21 3*15*27
%e 3*15*15 7*11*15 3*3*135
%e 3*3*3*5*5 3*3*3*5*9
%e 3*3*3*3*15
%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=0, 0, g(2*n+1$2, 1)):
%p seq(a(n), n=0..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[#]]&&OddQ[Times@@#]&]],{n,1,100,2}];
%Y The version for partitions is A160786, ranked by A300272.
%Y The not necessarily odd-length version is A340101.
%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 A316439 counts factorizations by product and length.
%Y Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
%K nonn
%O 0,14
%A _Gus Wiseman_, Dec 30 2020