A340101 - OEIS

%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