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A340689 Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.

%I A340689 #14 Feb 01 2021 14:16:18
%S A340689 1,16,384,576,864,1296,1944,2916,4374,6561,131072,196608,262144,
%T A340689 294912,393216,442368,524288,589824,663552,786432,884736,995328,
%U A340689 1048576,1179648,1327104,1492992,1572864,1769472,1990656,2097152,2239488,2359296,2654208,2985984,3145728
%N A340689 Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.
%H A340689 Chai Wah Wu, <a href="/A340689/b340689.txt">Table of n, a(n) for n = 1..10000</a>
%e A340689 The initial terms and a valid factorization of each are:
%e A340689          1 =
%e A340689         16 = 2*2*2*2
%e A340689        384 = 2*2*2*2*2*2*2*3
%e A340689        576 = 2*2*2*2*2*2*3*3
%e A340689        864 = 2*2*2*2*2*3*3*3
%e A340689       1296 = 2*2*2*2*3*3*3*3
%e A340689       1944 = 2*2*2*3*3*3*3*3
%e A340689       2916 = 2*2*3*3*3*3*3*3
%e A340689       4374 = 2*3*3*3*3*3*3*3
%e A340689       6561 = 3*3*3*3*3*3*3*3
%e A340689     131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
%e A340689     196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
%e A340689     262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
%e A340689     294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4
%t A340689 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340689 Select[Range[1000],Select[facs[#],Length[#]==2^Max@@#&]!={}&]
%Y A340689 Partitions of the prescribed type are counted by A340611.
%Y A340689 The conjugate version is A340690.
%Y A340689 A001055 counts factorizations, with strict case A045778.
%Y A340689 A047993 counts balanced partitions.
%Y A340689 A316439 counts factorizations by product and length.
%Y A340689 A340596 counts co-balanced factorizations.
%Y A340689 A340597 lists numbers with an alt-balanced factorization.
%Y A340689 A340653 counts balanced factorizations.
%Y A340689 Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.
%K A340689 nonn
%O A340689 1,2
%A A340689 _Gus Wiseman_, Jan 28 2021
%E A340689 More terms from _Chai Wah Wu_, Feb 01 2021