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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.
1
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
1,1
FORMULA
T(n,1) = 2^n.
EXAMPLE
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.
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]]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Paolo Xausa, Oct 03 2024
STATUS
approved