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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. 1

%I #31 Feb 06 2019 16:18:26

%S 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 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 16,3,22,28,5,1,1,2,3,10,6,4,3,22,14,30,36

%N 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 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 Alois P. Heinz, <a href="/A323376/b323376.txt">Antidiagonals n = 1..200, flattened</a>

%F T(n,k) = A250211(prime(n), prime(k)).

%e Table begins

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

%e n | p() | 2 3 5 7 11 13 17 19 23 29 ...

%e ---+-----+----------------------------------------

%e 1 | 2 | 0, 2, 4, 3, 10, 12, 8, 18, 11, 28, ...

%e 2 | 3 | 1, 0, 4, 6, 5, 3, 16, 18, 11, 28, ...

%e 3 | 5 | 1, 2, 0, 6, 5, 4, 16, 9, 22, 14, ...

%e 4 | 7 | 1, 1, 4, 0, 10, 12, 16, 3, 22, 7, ...

%e 5 | 11 | 1, 2, 1, 3, 0, 12, 16, 3, 22, 28, ...

%e 6 | 13 | 1, 1, 4, 2, 10, 0, 4, 18, 11, 14, ...

%e 7 | 17 | 1, 2, 4, 6, 10, 6, 0, 9, 22, 4, ...

%e 8 | 19 | 1, 1, 2, 6, 10, 12, 8, 0, 22, 28, ...

%e 9 | 23 | 1, 2, 4, 3, 1, 6, 16, 9 , 0, 7, ...

%e 10 | 29 | 1, 2, 2, 1, 10, 3, 16, 18, 11, 0, ...

%e ...

%p A:= (n, k)-> `if`(n=k, 0, (p-> numtheory[order](p(n), p(k)))(ithprime)):

%p seq(seq(A(1+d-k, k), k=1..d), d=1..14); # _Alois P. Heinz_, Feb 06 2019

%t 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 (PARI) T(n,k) = if(n==k, 0, znorder(Mod(prime(n), prime(k))))

%Y Cf. A250211.

%Y Cf. A014664 (1st row), A062117 (2nd row), A211241 (3rd row), A211243 (4th row), A039701 (2nd column).

%Y Cf. A226367 (lower diagonal), A226295 (upper diagonal).

%K nonn,tabl

%O 1,3

%A _Jianing Song_, Jan 12 2019

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Last modified March 29 05:28 EDT 2024. Contains 371264 sequences. (Running on oeis4.)