

A340597


Numbers with an altbalanced factorization.


17



4, 12, 18, 27, 32, 48, 64, 72, 80, 96, 108, 120, 128, 144, 160, 180, 192, 200, 240, 256, 270, 288, 300, 320, 360, 384, 400, 405, 432, 448, 450, 480, 500, 540, 576, 600, 640, 648, 672, 675, 720, 750, 768, 800, 864, 896, 900, 960, 972, 1000, 1008, 1024, 1080
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OFFSET

1,1


COMMENTS

We define a factorization into factors > 1 to be altbalanced if its length is equal to its greatest factor.


LINKS



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 altbalanced 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 twicebalanced version is A340657.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balancerelated sequences:
 A010054 counts balanced strict partitions.
 A047993 counts balanced partitions.
 A098124 counts balanced compositions.
 A106529 lists Heinz numbers of balanced partitions.
 A340596 counts cobalanced factorizations.
 A340598 counts balanced set partitions.
 A340600 counts unlabeled balanced multiset partitions.
 A340653 counts balanced factorizations.
 A340654 counts crossbalanced factorizations.
 A340655 counts twicebalanced factorizations.
Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656.


KEYWORD

nonn


AUTHOR



STATUS

approved



