A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset

0,14

Links

Table of n, a(n) for n=0..86.

Example

The factorizations for 2n + 1 = 135, 225, 315, 405, 675, 1155, 1215:
  135      225      315      405         675         1155      1215
  3*5*9    5*5*9    5*7*9    5*9*9       3*3*75      3*5*77    3*5*81
  3*3*15   3*3*25   3*3*35   3*3*45      3*5*45      3*7*55    3*9*45
           3*5*15   3*5*21   3*5*27      3*9*25      5*7*33    5*9*27
                    3*7*15   3*9*15      5*5*27      3*11*35   9*9*15
                             3*3*3*3*5   5*9*15      5*11*21   3*15*27
                                         3*15*15     7*11*15   3*3*135
                                         3*3*3*5*5             3*3*3*5*9
                                                               3*3*3*3*15

Maple

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=0, 0, g(2*n+1$2, 1)):
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[Length[#]]&&OddQ[Times@@#]&]], {n, 1, 100, 2}];

Crossrefs

The version for partitions is A160786, ranked by A300272.

The not necessarily odd-length version is A340101.

A000009 counts partitions into odd parts, ranked by A066208.

A001055 counts factorizations, with strict case A045778.

A027193 counts partitions of odd length, ranked by A026424.

A058695 counts partitions of odd numbers, ranked by A300063.

A316439 counts factorizations by product and length.

Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.

Sequence in context: A043282 A031226 A031242 * A031260 A360157 A298735

Adjacent sequences: A340099 A340100 A340101 * A340103 A340104 A340105

Keyword

nonn

Author

Gus Wiseman, Dec 30 2020

Status

approved