OFFSET
1,1
COMMENTS
We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor.
EXAMPLE
The sequence of terms together with their prime signatures begins:
4: (2) 180: (2,2,1) 450: (1,2,2)
12: (2,1) 192: (6,1) 480: (5,1,1)
18: (1,2) 200: (3,2) 500: (2,3)
27: (3) 240: (4,1,1) 540: (2,3,1)
32: (5) 256: (8) 576: (6,2)
48: (4,1) 270: (1,3,1) 600: (3,1,2)
64: (6) 288: (5,2) 640: (7,1)
72: (3,2) 300: (2,1,2) 648: (3,4)
80: (4,1) 320: (6,1) 672: (5,1,1)
96: (5,1) 360: (3,2,1) 675: (3,2)
108: (2,3) 384: (7,1) 720: (4,2,1)
120: (3,1,1) 400: (4,2) 750: (1,1,3)
128: (7) 405: (4,1) 768: (8,1)
144: (4,2) 432: (4,3) 800: (5,2)
160: (5,1) 448: (6,1) 864: (5,3)
For example, there are two alt-balanced factorizations of 480, namely (2*3*4*4*5) and (2*2*2*2*5*6), so 480 in the sequence.
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Select[facs[#], Length[#]==Max[#]&]!={}&]
CROSSREFS
Numbers with a balanced factorization are A100959.
These factorizations are counted by A340599.
The twice-balanced version is A340657.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
- A340655 counts twice-balanced factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2021
STATUS
approved