A376738 - OEIS

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.

%I #12 Oct 05 2024 00:24:28

%S 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,

%T 27,15,17,256,192,144,80,54,28,21,19,512,384,288,160,108,56,30,25,23,

%U 1024,768,576,320,216,112,60,42,35,29,2048,1536,1152,640,432,224,120,81,45,49,31

%N 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.

%F T(n,1) = 2^n.

%e Array begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10 ...

%e -----------------------------------------------------------------------

%e 1 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... = A000040

%e 2 | 4, 6, 9, 10, 14, 15, 21, 25, 35, 49, ... = A078972

%e 3 | 8, 12, 18, 20, 27, 28, 30, 42, 45, 50, ... = A376703

%e 4 | 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, ... = A376704

%e 5 | 32, 48, 72, 80, 108, 112, 120, 162, 168, 180, ...

%e 6 | 64, 96, 144, 160, 216, 224, 240, 324, 336, 360, ...

%e 7 | 128, 192, 288, 320, 432, 448, 480, 648, 672, 720, ...

%e 8 | 256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1440, ...

%e 9 | 512, 768, 1152, 1280, 1728, 1792, 1920, 2592, 2688, 2880, ...

%e 10 | 1024, 1536, 2304, 2560, 3456, 3584, 3840, 5184, 5376, 5760, ...

%e ... | \______ A376739 (main diagonal)

%e A000079 (from n = 1)

%e 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.

%t 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]]

%Y Cf. A000040, A000079, A078972, A376703, A376704, A376739.

%K nonn,base

%O 1,1

%A _Paolo Xausa_, Oct 03 2024