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A323376 Square array read by ascending antidiagonals: T(n,k) is the multiplicative order of the n-th prime modulo the k-th prime, or 0 if n = k, n >= 1, k >= 1.

%I A323376 #31 Feb 06 2019 16:18:26
%S A323376 0,1,2,1,0,4,1,2,4,3,1,1,0,6,10,1,2,4,6,5,12,1,1,1,0,5,3,8,1,2,4,3,10,
%T A323376 4,16,18,1,1,4,2,0,12,16,18,11,1,2,2,6,10,12,16,9,11,28,1,2,4,6,10,0,
%U A323376 16,3,22,28,5,1,1,2,3,10,6,4,3,22,14,30,36
%N A323376 Square array read by ascending antidiagonals: T(n,k) is the multiplicative order of the n-th prime modulo the k-th prime, or 0 if n = k, n >= 1, k >= 1.
%C A323376 The maximum element in the k-th column is prime(k) - 1. By Dirichlet's theorem on arithmetic progressions, all divisors of prime(k) - 1 occur infinitely many times in the n-th column.
%H A323376 Alois P. Heinz, <a href="/A323376/b323376.txt">Antidiagonals n = 1..200, flattened</a>
%F A323376 T(n,k) = A250211(prime(n), prime(k)).
%e A323376 Table begins
%e A323376      |  k  | 1  2  3  4   5   6   7   8   9  10  ...
%e A323376    n | p() | 2  3  5  7  11  13  17  19  23  29  ...
%e A323376   ---+-----+----------------------------------------
%e A323376    1 |   2 | 0, 2, 4, 3, 10, 12,  8, 18, 11, 28, ...
%e A323376    2 |   3 | 1, 0, 4, 6,  5,  3, 16, 18, 11, 28, ...
%e A323376    3 |   5 | 1, 2, 0, 6,  5,  4, 16,  9, 22, 14, ...
%e A323376    4 |   7 | 1, 1, 4, 0, 10, 12, 16,  3, 22,  7, ...
%e A323376    5 |  11 | 1, 2, 1, 3,  0, 12, 16,  3, 22, 28, ...
%e A323376    6 |  13 | 1, 1, 4, 2, 10,  0,  4, 18, 11, 14, ...
%e A323376    7 |  17 | 1, 2, 4, 6, 10,  6,  0,  9, 22,  4, ...
%e A323376    8 |  19 | 1, 1, 2, 6, 10, 12,  8,  0, 22, 28, ...
%e A323376    9 |  23 | 1, 2, 4, 3,  1,  6, 16,  9 , 0,  7, ...
%e A323376   10 |  29 | 1, 2, 2, 1, 10,  3, 16, 18, 11,  0, ...
%e A323376   ...
%p A323376 A:= (n, k)-> `if`(n=k, 0, (p-> numtheory[order](p(n), p(k)))(ithprime)):
%p A323376 seq(seq(A(1+d-k, k), k=1..d), d=1..14);  # _Alois P. Heinz_, Feb 06 2019
%t A323376 T[n_, k_] := If[n == k, 0, MultiplicativeOrder[Prime[n], Prime[k]]];Table[T[n, k], {n, 1, 10}, {k, 1, 10}] (* _Peter Luschny_, Jan 20 2019 *)
%o A323376 (PARI) T(n,k) = if(n==k, 0, znorder(Mod(prime(n), prime(k))))
%Y A323376 Cf. A250211.
%Y A323376 Cf. A014664 (1st row), A062117 (2nd row), A211241 (3rd row), A211243 (4th row), A039701 (2nd column).
%Y A323376 Cf. A226367 (lower diagonal), A226295 (upper diagonal).
%K A323376 nonn,tabl
%O A323376 1,3
%A A323376 _Jianing Song_, Jan 12 2019