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A323376
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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.
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1
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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
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,k) = A250211(prime(n), prime(k)).
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EXAMPLE
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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, ...
...
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MAPLE
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A:= (n, k)-> `if`(n=k, 0, (p-> numtheory[order](p(n), p(k)))(ithprime)):
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MATHEMATICA
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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 *)
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PROG
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(PARI) T(n, k) = if(n==k, 0, znorder(Mod(prime(n), prime(k))))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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