A340853 - OEIS

%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