cp's OEIS Frontend

A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

%I A340102 #7 Dec 30 2020 19:58:03
%S A340102 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 A340102 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 A340102 1,1,1,1,1,2,1,1,2,1,1,1,1,1,2,1,1,2,1
%N A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.
%e A340102 The factorizations for 2n + 1 = 135, 225, 315, 405, 675, 1155, 1215:
%e A340102   135      225      315      405         675         1155      1215
%e A340102   3*5*9    5*5*9    5*7*9    5*9*9       3*3*75      3*5*77    3*5*81
%e A340102   3*3*15   3*3*25   3*3*35   3*3*45      3*5*45      3*7*55    3*9*45
%e A340102            3*5*15   3*5*21   3*5*27      3*9*25      5*7*33    5*9*27
%e A340102                     3*7*15   3*9*15      5*5*27      3*11*35   9*9*15
%e A340102                              3*3*3*3*5   5*9*15      5*11*21   3*15*27
%e A340102                                          3*15*15     7*11*15   3*3*135
%e A340102                                          3*3*3*5*5             3*3*3*5*9
%e A340102                                                                3*3*3*3*15
%p A340102 g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
%p A340102       `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
%p A340102           d=numtheory[divisors](n) minus {1, n}))
%p A340102     end:
%p A340102 a:= n-> `if`(n=0, 0, g(2*n+1$2, 1)):
%p A340102 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 30 2020
%t A340102 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340102 Table[Length[Select[facs[n],OddQ[Length[#]]&&OddQ[Times@@#]&]],{n,1,100,2}];
%Y A340102 The version for partitions is A160786, ranked by A300272.
%Y A340102 The not necessarily odd-length version is A340101.
%Y A340102 A000009 counts partitions into odd parts, ranked by A066208.
%Y A340102 A001055 counts factorizations, with strict case A045778.
%Y A340102 A027193 counts partitions of odd length, ranked by A026424.
%Y A340102 A058695 counts partitions of odd numbers, ranked by A300063.
%Y A340102 A316439 counts factorizations by product and length.
%Y A340102 Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
%K A340102 nonn
%O A340102 0,14
%A A340102 _Gus Wiseman_, Dec 30 2020