A250211 -id:A250211 - cp's OEIS frontend

A139366 Table with the order r=r(N,n) of n modulo N, for given N and n, with gcd(N,n)=1.

0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 4, 4, 2, 0, 1, 0, 0, 0, 2, 0, 1, 3, 6, 3, 6, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 6, 0, 3, 6, 0, 3, 2, 0, 1, 0, 4, 0, 0, 0, 4, 0, 2, 0, 1, 10, 5, 5, 5, 10, 10, 10, 5, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 0, 1, 0, 6, 0, 6, 0, 0, 0, 3, 0

Offset: 1

Wolfdieter Lang, May 21 2008

In the table a 0 appears for 1 <= n <= N if gcd(N,n) is not 1. In particular, this is the case for the main diagonal with N > 1. Also for N=n=1 one sets r=0 because 1^m congruent to 0 (mod 1) for all m.
For given N and n with gcd(N,n)=1 the function F(N,n;a):=n^a (mod N) has period r=r(N,n): F(N,n;a+r) congruent F(N,n;a) (mod N).
The period r is used for factoring integers in quantum computing. See e.g. the Ekert and Jozsa reference.

			Triangle begins:
  [0];
  [1,0];
  [1,2,0];
  [1,0,2,0];
  [1,4,4,2,0];
  ...
For N=5, the order r of 3 (mod 5) is 4 because 3^1 == 3 (mod 5), 3^2 == 4 (mod 5), 3^3 == 2 (mod 5), 3^4 == 1 (mod 5). Hence F(5,3;a+4) == F(5,3;a) (mod 5).

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
A. Ekert and R. Josza, Quantum computation and Shor's factoring algorithm, Rev. Mod. Phys. 68 (1996) 733-753, sect. IV and Appendix A.
Wolfdieter Lang, First 15 rows and more.

Cf. A036391 (row sums).

See A250211 for another version.

a139366 1 1               = 0
a139366 n k | gcd n k > 1 = 0
            | otherwise   = head [r | r <- [1..], k ^ r `mod` n == 1]
a139366_row n = map (a139366 n) [1..n]
a139366_tabl = map a139366_row [1..]
-- Reinhard Zumkeller, May 01 2013

r[n_, k_] := If[ CoprimeQ[k, n], MultiplicativeOrder[k, n], 0]; Table[r[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

r(N,n)=if(N<2||gcd(n,N)>1,0,znorder(Mod(n,N)))
for(N=1,9,for(n=1,N,print1(r(N,n)", "))) \\ Charles R Greathouse IV, Feb 18 2013

r(N,n) is the smallest positive number with n^r == 1 (mod N), n=1..N, if gcd(N,n)=1, otherwise 0. This r is called the order of n (mod N) if gcd(N,n)=1.

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

A139366 Table with the order r=r(N,n) of n modulo N, for given N and n, with gcd(N,n)=1.

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