A340785 Number of factorizations of 2n into even factors > 1.

1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 3, 1, 11, 1, 2, 1, 6, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 12, 1, 3, 1, 4, 1, 3, 1, 7, 1, 2, 1, 7, 1, 2, 1, 15, 1, 3, 1, 4, 1, 3, 1, 12, 1, 2, 1, 4, 1, 3, 1, 12, 1, 2, 1, 7, 1

Offset: 1

Author

Gus Wiseman, Jan 30 2021

Keywords

• nonn

Examples

			The a(n) factorizations for n = 2*2, 2*4, 2*8, 2*12, 2*16, 2*32, 2*36, 2*48 are:
  4    8      16       24     32         64           72      96
  2*2  2*4    2*8      4*6    4*8        8*8          2*36    2*48
       2*2*2  4*4      2*12   2*16       2*32         4*18    4*24
              2*2*4    2*2*6  2*2*8      4*16         6*12    6*16
              2*2*2*2         2*4*4      2*4*8        2*6*6   8*12
                              2*2*2*4    4*4*4        2*2*18  2*6*8
                              2*2*2*2*2  2*2*16               4*4*6
                                         2*2*2*8              2*2*24
                                         2*2*4*4              2*4*12
                                         2*2*2*2*4            2*2*4*6
                                         2*2*2*2*2*2          2*2*2*12
                                                              2*2*2*2*6
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

The version for partitions is A035363 (A066207).

The odd version is A340101.

The even length case is A340786.

- Factorizations -

A001055 counts factorizations, with strict case A045778.

A340653 counts balanced factorizations.

A340831/A340832 count factorizations with odd maximum/minimum.

A316439 counts factorizations by product and length

A340102 counts odd-length factorizations of odd numbers into odd factors.

- Even -

A027187 counts partitions of even length/maximum (A028260/A244990).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A236913 counts partitions of even length and sum.

A340601 counts partitions of even rank (A340602).

Cf. A001147, A001222, A050320, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

Even bisection of A349906.

Programs

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],Select[#,OddQ]=={}&]],{n,2,100,2}]

PARI

A349906(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A349906(n/d, d))); (s));
A340785(n) = A349906(2*n); \\ Antti Karttunen, Dec 13 2021

Formula

a(n) = A349906(2*n). - Antti Karttunen, Dec 13 2021