A339890 Number of odd-length factorizations of n into factors > 1.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 5, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 2, 1, 1, 5, 1, 1, 1

Offset

1,8

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

a(n) + A339846(n) = A001055(n).

Example

The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
  24      48          60       72          96          120
  2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
  2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
          3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
          2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
          2*2*2*2*3            2*2*18      2*3*16      2*4*15
                               2*3*12      2*4*12      2*5*12
                               2*2*2*3*3   2*2*2*2*6   2*6*10
                                           2*2*2*3*4   3*4*10
                                                       2*2*2*3*5

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<2, 0, g(n$2, 1)):
seq(a(n), n=1..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[#]&]], {n, 100}]

Crossrefs

The case of set partitions (or n squarefree) is A024429.

The case of partitions (or prime powers) is A027193.

The ordered version is A174726 (even: A174725).

The remaining (even-length) factorizations are counted by A339846.

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.

A160786 counts odd-length partitions of odd numbers, ranked by A300272.

A316439 counts factorizations by product and length.

A340101 counts factorizations into odd factors.

A340102 counts odd-length factorizations into odd factors.

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

Sequence in context: A318749 A347708 A050330 * A351408 A205788 A214054

Adjacent sequences: A339887 A339888 A339889 * A339891 A339892 A339893

Keyword

nonn

Author

Gus Wiseman, Dec 28 2020

Status

approved