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

1,2

Links

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

Formula

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

Example

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

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}]

Prog

(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

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.

Sequence in context: A105609 A268361 A330569 * A355758 A206778 A101872

Adjacent sequences: A340782 A340783 A340784 * A340786 A340787 A340788

Keyword

nonn

Author

Gus Wiseman, Jan 30 2021

Status

approved