# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a340101
Showing 1-1 of 1

%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

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE