A376738 Array read by ascending antidiagonals: T(n,k) is the k-th number which is the product of n (possibly non-distinct) primes having the same number of decimal digits.
2, 4, 3, 8, 6, 5, 16, 12, 9, 7, 32, 24, 18, 10, 11, 64, 48, 36, 20, 14, 13, 128, 96, 72, 40, 27, 15, 17, 256, 192, 144, 80, 54, 28, 21, 19, 512, 384, 288, 160, 108, 56, 30, 25, 23, 1024, 768, 576, 320, 216, 112, 60, 42, 35, 29, 2048, 1536, 1152, 640, 432, 224, 120, 81, 45, 49, 31
Offset: 1
Examples
Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------------------
1 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... = A000040
2 | 4, 6, 9, 10, 14, 15, 21, 25, 35, 49, ... = A078972
3 | 8, 12, 18, 20, 27, 28, 30, 42, 45, 50, ... = A376703
4 | 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, ... = A376704
5 | 32, 48, 72, 80, 108, 112, 120, 162, 168, 180, ...
6 | 64, 96, 144, 160, 216, 224, 240, 324, 336, 360, ...
7 | 128, 192, 288, 320, 432, 448, 480, 648, 672, 720, ...
8 | 256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1440, ...
9 | 512, 768, 1152, 1280, 1728, 1792, 1920, 2592, 2688, 2880, ...
10 | 1024, 1536, 2304, 2560, 3456, 3584, 3840, 5184, 5376, 5760, ...
... | \______ A376739 (main diagonal)
A000079 (from n = 1)
T(9,5) = 1728 because 1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 is the 5th number with nine prime factors all having the same number of digits.
Programs
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Mathematica
Module[{dmax = 15, a, m, f}, a = Table[m = 2^n - 1; Table[While[Total[(f = FactorInteger[++m])[[All, 2]]] != n || Length[Union[IntegerLength[f[[All, 1]]]]] > 1]; m, dmax - n + 1], {n, dmax, 1, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]
Formula
T(n,1) = 2^n.