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Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.
4

%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