

A323376


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


1



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

1,3


COMMENTS

The maximum element in the kth column is prime(k)  1. By Dirichlet's theorem on arithmetic progressions, all divisors of prime(k)  1 occur infinitely many times in the nth column.


LINKS



FORMULA

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


EXAMPLE

Table begins
 k  1 2 3 4 5 6 7 8 9 10 ...
n  p()  2 3 5 7 11 13 17 19 23 29 ...
++
1  2  0, 2, 4, 3, 10, 12, 8, 18, 11, 28, ...
2  3  1, 0, 4, 6, 5, 3, 16, 18, 11, 28, ...
3  5  1, 2, 0, 6, 5, 4, 16, 9, 22, 14, ...
4  7  1, 1, 4, 0, 10, 12, 16, 3, 22, 7, ...
5  11  1, 2, 1, 3, 0, 12, 16, 3, 22, 28, ...
6  13  1, 1, 4, 2, 10, 0, 4, 18, 11, 14, ...
7  17  1, 2, 4, 6, 10, 6, 0, 9, 22, 4, ...
8  19  1, 1, 2, 6, 10, 12, 8, 0, 22, 28, ...
9  23  1, 2, 4, 3, 1, 6, 16, 9 , 0, 7, ...
10  29  1, 2, 2, 1, 10, 3, 16, 18, 11, 0, ...
...


MAPLE

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


MATHEMATICA

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 *)


PROG

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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



