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

Author

Gus Wiseman, Oct 17 2022

Keywords

• nonn

Examples

			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)
		

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.

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