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Number of factorizations of 2n + 1 into an odd number of odd factors > 1.
24

%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