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A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

%I A340690 #5 Jan 30 2021 22:51:25
%S A340690 2,8,12,16,32,48,64,72,80,96,112,120,128,144,160,168,192,200,224,240,
%T A340690 256,280,288,320,336,384,392,432,448,480,512,576,640,672,704,720,768,
%U A340690 800,832,864,896,960,1008,1024,1056,1120,1152,1200,1248,1280,1296,1344
%N A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.
%e A340690 The initial terms and a valid factorization of each:
%e A340690       2 = 2           168 = 3*7*8        512 = 2*2*2*2*32
%e A340690       8 = 2*4         192 = 2*2*3*16     576 = 2*2*9*16
%e A340690      12 = 3*4         200 = 5*5*8        640 = 2*2*10*16
%e A340690      16 = 4*4         224 = 4*7*8        672 = 2*3*7*16
%e A340690      32 = 2*2*8       240 = 5*6*8        704 = 2*2*11*16
%e A340690      48 = 2*3*8       256 = 2*2*4*16     720 = 3*3*5*16
%e A340690      64 = 2*4*8       280 = 5*7*8        768 = 2*2*2*3*32
%e A340690      72 = 3*3*8       288 = 2*3*3*16     800 = 2*5*5*16
%e A340690      80 = 2*5*8       320 = 2*2*5*16     832 = 2*2*13*16
%e A340690      96 = 2*6*8       336 = 6*7*8        864 = 2*3*9*16
%e A340690     112 = 2*7*8       384 = 2*2*6*16     896 = 2*2*14*16
%e A340690     120 = 3*5*8       392 = 7*7*8        960 = 2*2*15*16
%e A340690     128 = 2*2*2*16    432 = 3*3*3*16    1008 = 3*3*7*16
%e A340690     144 = 3*6*8       448 = 2*2*7*16    1024 = 2*2*2*4*32
%e A340690     160 = 4*5*8       480 = 2*3*5*16    1056 = 2*3*11*16
%t A340690 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340690 Select[Range[1000],Select[facs[#],2^Length[#]==Max@@#&]!={}&]
%Y A340690 Partitions of the prescribed type are counted by A340611.
%Y A340690 The conjugate version is A340689.
%Y A340690 A001055 counts factorizations, with strict case A045778.
%Y A340690 A047993 counts balanced partitions.
%Y A340690 A316439 counts factorizations by product and length.
%Y A340690 A340596 counts co-balanced factorizations.
%Y A340690 A340597 lists numbers with an alt-balanced factorization.
%Y A340690 A340653 counts balanced factorizations.
%Y A340690 Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.
%K A340690 nonn
%O A340690 1,1
%A A340690 _Gus Wiseman_, Jan 28 2021