OFFSET
1,1
EXAMPLE
The initial terms and a valid factorization of each:
2 = 2 168 = 3*7*8 512 = 2*2*2*2*32
8 = 2*4 192 = 2*2*3*16 576 = 2*2*9*16
12 = 3*4 200 = 5*5*8 640 = 2*2*10*16
16 = 4*4 224 = 4*7*8 672 = 2*3*7*16
32 = 2*2*8 240 = 5*6*8 704 = 2*2*11*16
48 = 2*3*8 256 = 2*2*4*16 720 = 3*3*5*16
64 = 2*4*8 280 = 5*7*8 768 = 2*2*2*3*32
72 = 3*3*8 288 = 2*3*3*16 800 = 2*5*5*16
80 = 2*5*8 320 = 2*2*5*16 832 = 2*2*13*16
96 = 2*6*8 336 = 6*7*8 864 = 2*3*9*16
112 = 2*7*8 384 = 2*2*6*16 896 = 2*2*14*16
120 = 3*5*8 392 = 7*7*8 960 = 2*2*15*16
128 = 2*2*2*16 432 = 3*3*3*16 1008 = 3*3*7*16
144 = 3*6*8 448 = 2*2*7*16 1024 = 2*2*2*4*32
160 = 4*5*8 480 = 2*3*5*16 1056 = 2*3*11*16
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[#], 2^Length[#]==Max@@#&]!={}&]
CROSSREFS
Partitions of the prescribed type are counted by A340611.
The conjugate version is A340689.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 28 2021
STATUS
approved