A248736 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.

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

Offset: 1

Author

Robert G. Wilson v, Oct 12 2014

Keywords

• nonn
• base
• tabl

Comments

This array is based upon the conjectured formula by Charles R Greathouse IV in A077644, adapted to other bases.

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.

Examples

			\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
		

Crossrefs

Cf. A129490 (row 2), A248729 (row 3), A248731 (row 5), A248733 (row 6), A248735 (row 7), A077644 (row 10).

Programs

Mathematica

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

Formula

a(b,n) = ceiling(Pi*sqrt(2/3)*sqrt(b)^n - log(48)/2 - n*log b) / log(10).