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

1,2

Links

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

Example

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

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@@#&]!={}&]

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.

Sequence in context: A189766 A235442 A343213 * A302262 A302962 A302805

Adjacent sequences: A340686 A340687 A340688 * A340690 A340691 A340692

Keyword

nonn

Author

Gus Wiseman, Jan 28 2021

Extensions

More terms from Chai Wah Wu, Feb 01 2021

Status

approved