A062117 -id:A062117 - cp's OEIS frontend

A240658 Least k such that 3^k == -1 (mod prime(n)), or 0 if no such k exists.

1, 0, 2, 3, 0, 0, 8, 9, 0, 14, 15, 9, 4, 21, 0, 26, 0, 5, 11, 0, 6, 39, 0, 44, 24, 50, 17, 0, 0, 56, 63, 0, 68, 69, 74, 25, 39, 81, 0, 86, 0, 0, 0, 8, 98, 99, 105, 111, 0, 0, 116, 0, 60, 0, 128, 0, 134, 15, 0, 140, 141, 146, 17, 0, 0, 158, 165, 84, 0, 87, 176, 0

Offset: 1

T. D. Noe, Apr 14 2014

nonn

The least k, if it exists, such that prime(n) divides 3^k + 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A062117 (order of 3 mod prime(n)).

Table[p = Prime[n]; s = Select[Range[p/2], PowerMod[3, #, p] == p - 1 &, 1]; If[s == {}, 0, s[[1]]], {n, 100}]

a(1) = 1; for n > 1, a(n) = A062117(n)/2 if A062117(n) is even, otherwise 0.

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.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Jan 12 2019

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.

			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, ...
  ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Cf. A250211.
Cf. A014664 (1st row), A062117 (2nd row), A211241 (3rd row), A211243 (4th row), A039701 (2nd column).
Cf. A226367 (lower diagonal), A226295 (upper diagonal).

A:= (n, k)-> `if`(n=k, 0, (p-> numtheory[order](p(n), p(k)))(ithprime)):
seq(seq(A(1+d-k, k), k=1..d), d=1..14);  # Alois P. Heinz, Feb 06 2019

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 *)

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

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

cp's OEIS Frontend

A240658 Least k such that 3^k == -1 (mod prime(n)), or 0 if no such k exists.

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