A340689 - OEIS

%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