This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340101 #15 Dec 14 2021 05:46:23
%S A340101 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 A340101 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 A340101 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 A340101 Number of factorizations of 2n + 1 into odd factors > 1.
%H A340101 Antti Karttunen, <a href="/A340101/b340101.txt">Table of n, a(n) for n = 0..32768</a>
%F A340101 a(n) = A001055(2n+1).
%F A340101 a(n) = A349907(2n+1). - _Antti Karttunen_, Dec 13 2021
%e A340101 The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
%e A340101 27 45 135 225 315 405 1155
%e A340101 3*9 5*9 3*45 3*75 5*63 5*81 15*77
%e A340101 3*3*3 3*15 5*27 5*45 7*45 9*45 21*55
%e A340101 3*3*5 9*15 9*25 9*35 15*27 33*35
%e A340101 3*5*9 15*15 15*21 3*135 3*385
%e A340101 3*3*15 5*5*9 3*105 5*9*9 5*231
%e A340101 3*3*3*5 3*3*25 5*7*9 3*3*45 7*165
%e A340101 3*5*15 3*3*35 3*5*27 11*105
%e A340101 3*3*5*5 3*5*21 3*9*15 3*5*77
%e A340101 3*7*15 3*3*5*9 3*7*55
%e A340101 3*3*5*7 3*3*3*15 5*7*33
%e A340101 3*3*3*3*5 3*11*35
%e A340101 5*11*21
%e A340101 7*11*15
%e A340101 3*5*7*11
%p A340101 g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
%p A340101 `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
%p A340101 d=numtheory[divisors](n) minus {1, n}))
%p A340101 end:
%p A340101 a:= n-> g(2*n+1$2):
%p A340101 seq(a(n), n=0..100); # _Alois P. Heinz_, Dec 30 2020
%t A340101 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340101 Table[Length[Select[facs[n],OddQ[Times@@#]&]],{n,1,100,2}]
%o A340101 (PARI)
%o A340101 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 A340101(n) = A001055(n+n+1); \\ _Antti Karttunen_, Dec 13 2021
%Y A340101 The version for partitions is A160786, ranked by A300272.
%Y A340101 The even version is A340785.
%Y A340101 The odd-length case is A340102.
%Y A340101 A000009 counts partitions into odd parts, ranked by A066208.
%Y A340101 A001055 counts factorizations, with strict case A045778.
%Y A340101 A027193 counts partitions of odd length, ranked by A026424.
%Y A340101 A058695 counts partitions of odd numbers, ranked by A300063.
%Y A340101 A316439 counts factorizations by product and length.
%Y A340101 Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
%Y A340101 Odd bisection of A001055, and also of A349907.
%K A340101 nonn
%O A340101 0,5
%A A340101 _Gus Wiseman_, Dec 28 2020
%E A340101 Data section extended up to 105 terms by _Antti Karttunen_, Dec 13 2021