cp's OEIS Frontend

A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

%I A340853 #9 Feb 04 2021 20:53:38
%S A340853 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 A340853 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 A340853 1,1,1,4,1,1,1,2,1,1,1,4,2,1,1,3,1,1,1
%N A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.
%C A340853 Also factorizations whose greatest common divisor is a multiple of the number of factors.
%e A340853 The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
%e A340853   2   4     16    48     96     144     216      240     432
%e A340853       2*2   2*8   6*8    2*48   2*72    4*54     4*60    6*72
%e A340853             4*4   2*24   4*24   4*36    6*36     6*40    8*54
%e A340853                   4*12   6*16   6*24    12*18    8*30    12*36
%e A340853                          8*12   8*18    2*108    10*24   18*24
%e A340853                                 12*12   6*6*6    12*20   2*216
%e A340853                                         3*3*24   2*120   4*108
%e A340853                                         3*6*12           3*3*48
%e A340853                                                          3*6*24
%e A340853                                                          6*6*12
%e A340853                                                          3*12*12
%t A340853 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340853 Table[Length[Select[facs[n],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]
%Y A340853 Positions of 1's are A048103.
%Y A340853 Positions of terms > 1 are A100716.
%Y A340853 The version for partitions is A143773 (A316428).
%Y A340853 The reciprocal for partitions is A340693 (A340606).
%Y A340853 The version for strict partitions is A340830.
%Y A340853 The reciprocal version is A340851.
%Y A340853 A320911 can be factored into squarefree semiprimes.
%Y A340853 A340597 have an alt-balanced factorization.
%Y A340853 A340656 lack a twice-balanced factorization, complement A340657.
%Y A340853 - Factorizations -
%Y A340853 A001055 counts factorizations, with strict case A045778.
%Y A340853 A316439 counts factorizations by product and length.
%Y A340853 A339846 counts factorizations of even length.
%Y A340853 A339890 counts factorizations of odd length.
%Y A340853 A340101 counts factorizations into odd factors, odd-length case A340102.
%Y A340853 A340653 counts balanced factorizations.
%Y A340853 A340785 counts factorizations into even factors, even-length case A340786.
%Y A340853 A340831/A340832 counts factorizations with odd maximum/minimum.
%Y A340853 A340854 cannot be factored with odd least factor, complement A340855.
%Y A340853 Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.
%K A340853 nonn
%O A340853 1,4
%A A340853 _Gus Wiseman_, Feb 04 2021