%I #6 Feb 04 2021 20:53:21

%S 1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of factorizations of n such that every factor is a divisor of the number of factors.

%C Also factorizations whose number of factors is divisible by their least common multiple.

%e The a(n) factorizations for n = 8192, 46656, 73728:

%e 2*2*2*2*2*4*8*8 6*6*6*6*6*6 2*2*2*2*2*2*2*2*2*4*6*6

%e 2*2*2*2*4*4*4*8 2*2*2*2*2*2*3*3*3*3*3*3 2*2*2*2*2*2*2*2*3*4*4*6

%e 2*2*2*4*4*4*4*4 2*2*2*2*2*2*2*3*3*4*4*4

%e 2*2*2*2*2*2*2*2*2*2*2*4 2*2*2*2*2*2*2*2*2*2*6*12

%e 2*2*2*2*2*2*2*2*2*3*4*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[n],And@@IntegerQ/@(Length[#]/#)&]],{n,100}]

%Y The version for partitions is A340693, with reciprocal version A143773.

%Y Positions of nonzero terms are A340852.

%Y The reciprocal version is A340853.

%Y A320911 can be factored into squarefree semiprimes.

%Y A340597 have an alt-balanced factorization.

%Y A340656 lack a twice-balanced factorization, complement A340657.

%Y - Factorizations -

%Y A001055 counts factorizations, with strict case A045778.

%Y A316439 counts factorizations by product and length.

%Y A339846 counts factorizations of even length.

%Y A339890 counts factorizations of odd length.

%Y A340101 counts factorizations into odd factors, odd-length case A340102.

%Y A340653 counts balanced factorizations.

%Y A340785 counts factorizations into even numbers, even-length case A340786.

%Y A340831/A340832 count factorizations with odd maximum/minimum.

%Y A340854 cannot be factored with odd least factor, complement A340855.

%Y Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

%K nonn

%O 1,64

%A _Gus Wiseman_, Feb 04 2021