A340101 Number of factorizations of 2n + 1 into odd factors > 1.
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, 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, 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
Offset: 0
Keywords
Examples
The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
27 45 135 225 315 405 1155
3*9 5*9 3*45 3*75 5*63 5*81 15*77
3*3*3 3*15 5*27 5*45 7*45 9*45 21*55
3*3*5 9*15 9*25 9*35 15*27 33*35
3*5*9 15*15 15*21 3*135 3*385
3*3*15 5*5*9 3*105 5*9*9 5*231
3*3*3*5 3*3*25 5*7*9 3*3*45 7*165
3*5*15 3*3*35 3*5*27 11*105
3*3*5*5 3*5*21 3*9*15 3*5*77
3*7*15 3*3*5*9 3*7*55
3*3*5*7 3*3*3*15 5*7*33
3*3*3*3*5 3*11*35
5*11*21
7*11*15
3*5*7*11
Links
- Antti Karttunen, Table of n, a(n) for n = 0..32768
Crossrefs
Programs
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Maple
g:= proc(n, k) option remember; `if`(n>k, 0, 1)+ `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)), d=numtheory[divisors](n) minus {1, n})) end: a:= n-> g(2*n+1$2): seq(a(n), n=0..100); # Alois P. Heinz, Dec 30 2020 -
Mathematica
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]]; Table[Length[Select[facs[n],OddQ[Times@@#]&]],{n,1,100,2}] -
PARI
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 A340101(n) = A001055(n+n+1); \\ Antti Karttunen, Dec 13 2021
Formula
a(n) = A001055(2n+1).
a(n) = A349907(2n+1). - Antti Karttunen, Dec 13 2021
Extensions
Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021
Comments