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%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
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