cp's OEIS Frontend

A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.

%I A340852 #9 Feb 04 2021 20:53:29
%S A340852 1,4,16,27,32,64,96,128,144,192,216,256,288,324,432,486,512,576,648,
%T A340852 729,864,972,1024,1296,1458,1728,1944,2048,2560,2592,2916,3125,3888,
%U A340852 4096,5120,5184,5832,6144,6400,7776,8192,9216,11664,12288,12800,13824,15552
%N A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.
%C A340852 Also numbers that can be factored in such a way that the length is divisible by the least common multiple.
%e A340852 The sequence of terms together with their prime indices begins:
%e A340852     1: {}
%e A340852     4: {1,1}
%e A340852    16: {1,1,1,1}
%e A340852    27: {2,2,2}
%e A340852    32: {1,1,1,1,1}
%e A340852    64: {1,1,1,1,1,1}
%e A340852    96: {1,1,1,1,1,2}
%e A340852   128: {1,1,1,1,1,1,1}
%e A340852   144: {1,1,1,1,2,2}
%e A340852   192: {1,1,1,1,1,1,2}
%e A340852   216: {1,1,1,2,2,2}
%e A340852   256: {1,1,1,1,1,1,1,1}
%e A340852   288: {1,1,1,1,1,2,2}
%e A340852   324: {1,1,2,2,2,2}
%e A340852   432: {1,1,1,1,2,2,2}
%e A340852 For example, 24576 has three suitable factorizations:
%e A340852   (2*2*2*2*2*2*2*2*2*2*2*12)
%e A340852   (2*2*2*2*2*2*2*2*2*2*4*6)
%e A340852   (2*2*2*2*2*2*2*2*2*3*4*4)
%e A340852 so is in the sequence.
%t A340852 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340852 Select[Range[1000],Select[facs[#],And@@IntegerQ/@(Length[#]/#)&]!={}&]
%Y A340852 Partitions of this type are counted by A340693 (A340606).
%Y A340852 These factorizations are counted by A340851.
%Y A340852 The reciprocal version is A340853.
%Y A340852 A143773 counts partitions whose parts are multiples of the number of parts.
%Y A340852 A320911 can be factored into squarefree semiprimes.
%Y A340852 A340597 have an alt-balanced factorization.
%Y A340852 A340656 lack a twice-balanced factorization, complement A340657.
%Y A340852 - Factorizations -
%Y A340852 A001055 counts factorizations, with strict case A045778.
%Y A340852 A316439 counts factorizations by product and length.
%Y A340852 A339846 counts factorizations of even length.
%Y A340852 A339890 counts factorizations of odd length.
%Y A340852 A340101 counts factorizations into odd factors, odd-length case A340102.
%Y A340852 A340653 counts balanced factorizations.
%Y A340852 A340831/A340832 count factorizations with odd maximum/minimum.
%Y A340852 A340785 counts factorizations into even numbers, even-length case A340786.
%Y A340852 A340854 cannot be factored with odd least factor, complement A340855.
%Y A340852 Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.
%K A340852 nonn
%O A340852 1,2
%A A340852 _Gus Wiseman_, Feb 04 2021