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%I #15 Dec 14 2021 05:46:23
%S 1,1,1,1,2,1,1,2,1,1,2,1,2,3,1,1,2,2,1,2,1,1,4,1,2,2,1,2,2,1,1,4,2,1,
%T 2,1,1,4,2,1,5,1,2,2,1,2,2,2,1,4,1,1,5,1,1,2,1,2,4,2,2,2,3,1,2,1,2,7,
%U 1,1,2,2,2,4,1,1,4,2,1,2,2,1,5,1,2,4,1,4,2,1,1,2,2,2,7,1,1,5,1,1,2,2,2,4,2
%N Number of factorizations of 2n + 1 into odd factors > 1.
%H Antti Karttunen, <a href="/A340101/b340101.txt">Table of n, a(n) for n = 0..32768</a>
%F a(n) = A001055(2n+1).
%F a(n) = A349907(2n+1). - _Antti Karttunen_, Dec 13 2021
%e The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
%e 27 45 135 225 315 405 1155
%e 3*9 5*9 3*45 3*75 5*63 5*81 15*77
%e 3*3*3 3*15 5*27 5*45 7*45 9*45 21*55
%e 3*3*5 9*15 9*25 9*35 15*27 33*35
%e 3*5*9 15*15 15*21 3*135 3*385
%e 3*3*15 5*5*9 3*105 5*9*9 5*231
%e 3*3*3*5 3*3*25 5*7*9 3*3*45 7*165
%e 3*5*15 3*3*35 3*5*27 11*105
%e 3*3*5*5 3*5*21 3*9*15 3*5*77
%e 3*7*15 3*3*5*9 3*7*55
%e 3*3*5*7 3*3*3*15 5*7*33
%e 3*3*3*3*5 3*11*35
%e 5*11*21
%e 7*11*15
%e 3*5*7*11
%p g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
%p `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
%p d=numtheory[divisors](n) minus {1, n}))
%p end:
%p a:= n-> g(2*n+1$2):
%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[Times@@#]&]],{n,1,100,2}]
%o (PARI)
%o A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
%o A340101(n) = A001055(n+n+1); \\ _Antti Karttunen_, Dec 13 2021
%Y The version for partitions is A160786, ranked by A300272.
%Y The even version is A340785.
%Y The odd-length case is A340102.
%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.
%Y Odd bisection of A001055, and also of A349907.
%K nonn
%O 0,5
%A _Gus Wiseman_, Dec 28 2020
%E Data section extended up to 105 terms by _Antti Karttunen_, Dec 13 2021
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