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Number of factorizations of n such that every factor is a multiple of the number of factors.
8

%I #9 Feb 04 2021 20:53:38

%S 0,1,1,2,1,1,1,2,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,3,1,1,2,2,1,1,1,3,1,1,

%T 1,3,1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,2,1,3,1,1,1,3,1,1,1,4,1,1,1,2,

%U 1,1,1,4,1,1,1,2,1,1,1,4,2,1,1,3,1,1,1

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

%C Also factorizations whose greatest common divisor is a multiple of the number of factors.

%e The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:

%e 2 4 16 48 96 144 216 240 432

%e 2*2 2*8 6*8 2*48 2*72 4*54 4*60 6*72

%e 4*4 2*24 4*24 4*36 6*36 6*40 8*54

%e 4*12 6*16 6*24 12*18 8*30 12*36

%e 8*12 8*18 2*108 10*24 18*24

%e 12*12 6*6*6 12*20 2*216

%e 3*3*24 2*120 4*108

%e 3*6*12 3*3*48

%e 3*6*24

%e 6*6*12

%e 3*12*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],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]

%Y Positions of 1's are A048103.

%Y Positions of terms > 1 are A100716.

%Y The version for partitions is A143773 (A316428).

%Y The reciprocal for partitions is A340693 (A340606).

%Y The version for strict partitions is A340830.

%Y The reciprocal version is A340851.

%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 factors, even-length case A340786.

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

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

%Y Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.

%K nonn

%O 1,4

%A _Gus Wiseman_, Feb 04 2021