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Number of integer factorizations of 2n into distinct even factors.
1

%I #6 Oct 17 2022 12:32:15

%S 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,

%T 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,

%U 1,3,1,7,1,2,1,3,1,3,1,7,1,2,1,6,1,2,1

%N Number of integer factorizations of 2n into distinct even factors.

%e The a(n) factorizations for n = 2, 4, 12, 24, 32, 48, 60, 96:

%e (4) (8) (24) (48) (64) (96) (120) (192)

%e (2*4) (4*6) (6*8) (2*32) (2*48) (2*60) (2*96)

%e (2*12) (2*24) (4*16) (4*24) (4*30) (4*48)

%e (4*12) (2*4*8) (6*16) (6*20) (6*32)

%e (2*4*6) (8*12) (10*12) (8*24)

%e (2*6*8) (2*6*10) (12*16)

%e (2*4*12) (4*6*8)

%e (2*4*24)

%e (2*6*16)

%e (2*8*12)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[2*n],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

%Y The version for partitions instead of factorizations is A000009.

%Y Positions of 1's are A004280.

%Y The non-strict version is A340785.

%Y Including odd n gives A357860.

%Y A000005 counts divisors.

%Y A001055 counts factorizations.

%Y A001221 counts distinct prime factors, sum A001414.

%Y A001222 counts prime-power divisors.

%Y A050361 counts strict factorizations into prime powers.

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 17 2022