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%I #14 Feb 01 2021 14:16:18
%S 1,16,384,576,864,1296,1944,2916,4374,6561,131072,196608,262144,
%T 294912,393216,442368,524288,589824,663552,786432,884736,995328,
%U 1048576,1179648,1327104,1492992,1572864,1769472,1990656,2097152,2239488,2359296,2654208,2985984,3145728
%N Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.
%H Chai Wah Wu, <a href="/A340689/b340689.txt">Table of n, a(n) for n = 1..10000</a>
%e The initial terms and a valid factorization of each are:
%e 1 =
%e 16 = 2*2*2*2
%e 384 = 2*2*2*2*2*2*2*3
%e 576 = 2*2*2*2*2*2*3*3
%e 864 = 2*2*2*2*2*3*3*3
%e 1296 = 2*2*2*2*3*3*3*3
%e 1944 = 2*2*2*3*3*3*3*3
%e 2916 = 2*2*3*3*3*3*3*3
%e 4374 = 2*3*3*3*3*3*3*3
%e 6561 = 3*3*3*3*3*3*3*3
%e 131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
%e 196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
%e 262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
%e 294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4
%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[#],Length[#]==2^Max@@#&]!={}&]
%Y Partitions of the prescribed type are counted by A340611.
%Y The conjugate version is A340690.
%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 1,2
%A _Gus Wiseman_, Jan 28 2021
%E More terms from _Chai Wah Wu_, Feb 01 2021
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