login
Number of factorizations of 2n + 1 into odd factors > 1.
28

%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