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A340597
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Numbers with an alt-balanced factorization.
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17
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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
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1,1
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COMMENTS
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We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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[#]&]!={}&]
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CROSSREFS
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Numbers with a balanced factorization are A100959.
These factorizations are counted by A340599.
The twice-balanced version is A340657.
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.
Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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