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

1, 16, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 131072, 196608, 262144, 294912, 393216, 442368, 524288, 589824, 663552, 786432, 884736, 995328, 1048576, 1179648, 1327104, 1492992, 1572864, 1769472, 1990656, 2097152, 2239488, 2359296, 2654208, 2985984, 3145728

Offset: 1

Author

Gus Wiseman, Jan 28 2021

Keywords

• nonn

Examples

			The initial terms and a valid factorization of each are:
         1 =
        16 = 2*2*2*2
       384 = 2*2*2*2*2*2*2*3
       576 = 2*2*2*2*2*2*3*3
       864 = 2*2*2*2*2*3*3*3
      1296 = 2*2*2*2*3*3*3*3
      1944 = 2*2*2*3*3*3*3*3
      2916 = 2*2*3*3*3*3*3*3
      4374 = 2*3*3*3*3*3*3*3
      6561 = 3*3*3*3*3*3*3*3
    131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
    196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
    262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
    294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Partitions of the prescribed type are counted by A340611.

The conjugate version is A340690.

A001055 counts factorizations, with strict case A045778.

A047993 counts balanced partitions.

A316439 counts factorizations by product and length.

A340596 counts co-balanced factorizations.

A340597 lists numbers with an alt-balanced factorization.

A340653 counts balanced factorizations.

Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],Length[#]==2^Max@@#&]!={}&]

Extensions

More terms from Chai Wah Wu, Feb 01 2021