A357859 Number of integer factorizations of 2n into distinct even factors.

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 4, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 6, 1, 2, 1, 5, 1, 3, 1, 3, 1, 3, 1, 7, 1, 2, 1, 3, 1, 3, 1, 7, 1, 2, 1, 6, 1, 2, 1

Offset

1,4

Links

Table of n, a(n) for n=1..87.

Example

The a(n) factorizations for n = 2, 4, 12, 24, 32, 48, 60, 96:
  (4)  (8)    (24)    (48)     (64)     (96)      (120)     (192)
       (2*4)  (4*6)   (6*8)    (2*32)   (2*48)    (2*60)    (2*96)
              (2*12)  (2*24)   (4*16)   (4*24)    (4*30)    (4*48)
                      (4*12)   (2*4*8)  (6*16)    (6*20)    (6*32)
                      (2*4*6)           (8*12)    (10*12)   (8*24)
                                        (2*6*8)   (2*6*10)  (12*16)
                                        (2*4*12)            (4*6*8)
                                                            (2*4*24)
                                                            (2*6*16)
                                                            (2*8*12)

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[2*n], UnsameQ@@#&&OddQ[Times@@(#+1)]&]], {n, 100}]

Crossrefs

The version for partitions instead of factorizations is A000009.

Positions of 1's are A004280.

The non-strict version is A340785.

Including odd n gives A357860.

A000005 counts divisors.

A001055 counts factorizations.

A001221 counts distinct prime factors, sum A001414.

A001222 counts prime-power divisors.

A050361 counts strict factorizations into prime powers.

Cf. A000688, A000961, A023894, A295935, A318721.

Sequence in context: A167679 A051881 A363852 * A338731 A081757 A323880

Adjacent sequences: A357856 A357857 A357858 * A357860 A357861 A357862

Keyword

nonn

Author

Gus Wiseman, Oct 17 2022

Status

approved