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A248736
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Array, read by antidiagonals, of the numbers of digits in the decimal expansion of the number of partitions of b^n employing a conjectured formula. See both the Comments and the Mathematica coding.
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4
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1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 2, 4, 7, 8, 4, 1, 2, 5, 10, 15, 15, 7, 1, 2, 6, 14, 25, 32, 27, 10, 1, 2, 7, 18, 37, 58, 67, 48, 15, 1, 2, 8, 22, 51, 94, 135, 138, 86, 22, 1, 2, 9, 27, 67, 140, 236, 306, 280, 152, 32, 1, 2, 10, 32, 86, 197, 377, 584, 690, 565, 266, 47
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OFFSET
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1,9
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COMMENTS
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As far as the direct computations for bases b = 2..12 and powers n=0..12 cited in cross references are concerned, the values computed here conform to the exact numbers of partitions.
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LINKS
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FORMULA
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a(b,n) = ceiling(Pi*sqrt(2/3)*sqrt(b)^n - log(48)/2 - n*log b) / log(10).
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EXAMPLE
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\n 0 1 2 3 4 5 6 7 8 9 10 11 ...
b\
2 1 1 1 2 3 4 7 10 15 22 32 47
3 1 1 2 4 8 15 27 48 86 152 266 463
4 1 1 3 7 15 32 67 138 280 565 1134 2275
5 1 1 4 10 25 58 135 306 690 1550 3474 7776
6 1 2 5 14 37 94 236 584 1437 3529 8654 21210
7 1 2 6 18 51 140 377 1005 2668 7069 18714 49527
8 1 2 7 22 67 197 565 1607 4555 12898 36494 103238
9 1 2 8 27 86 266 806 2429 7301 21918 65771 197332
10 1 2 9 32 107 347 1108 3515 11132 35219 111391 352269
11 1 2 10 37 130 442 1476 4910 16302 54085 179401 595031
12 1 2 11 43 156 550 1918 6661 23091 80011 277190 960240
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MATHEMATICA
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f[n_, b_] := Ceiling[(Pi*Sqrt[2/3]*Sqrt[b]^n - Log[48]/2 - n*Log[b]) / Log[10]]; Table[ f[n - b, b], {n, 2, 20}, {b, n, 2, -1}] // Flatten
(* cross checked with *) g[n_, b_] := f[n, b] = Floor[ Log10[ PartitionsP[ b^n]] + 1]; Table[ f[n - b, b], {n, 2, 20}, {b, n, 2, -1}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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