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