A124223 Table T(n,k) = reciprocal of k modulo prime(n), for 1 <= k < prime(n), read by rows.

1, 1, 2, 1, 3, 2, 4, 1, 4, 5, 2, 3, 6, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12, 1, 9, 6, 13, 7, 3, 5, 15, 2, 12, 14, 10, 4, 11, 8, 16, 1, 10, 13, 5, 4, 16, 11, 12, 17, 2, 7, 8, 3, 15, 14, 6, 9, 18, 1, 12, 8, 6, 14, 4, 10, 3, 18, 7, 21, 2, 16, 5, 20, 13, 19, 9, 17, 15, 11, 22

Offset: 1

Author

Franklin T. Adams-Watters, Oct 20 2006

Keywords

• nonn
• tabf

Comments

T(n,k) = smallest m such that k*m == 1 (mod prime(n)); prime(n) is the n-th prime: A000040(n).

Examples

			From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)
Table begins:
  1;
  1,2;
  1,3,2,4;
  1,4,5,2,3,6;
  1,6,4,3,9,2,8,7,5,10;
  1,7,9,10,8,11,2,5,3,4,6,12;
  1,9,6,13,7,3,5,15,2,12,14,10,4,11,8,16;
  1,10,13,5,4,16,11,12,17,2,7,8,3,15,14,6,9,18;
  1,12,8,6,14,4,10,3,18,7,21,2,16,5,20,13,19,9,17,15,11,22;
  ... (End)
		

Links

• Franklin T. Adams-Watters, Table of n, a(n) for n = 1..4181 (primes less than 200)
• Eric Weisstein's World of Mathematics, Modular Inverse

Crossrefs

Cf. A124224, A102057, A000040, A006093 (row lengths).

Programs

Maple

seq(seq(k^(ithprime(n)-2) mod ithprime(n), k=1..ithprime(n)-1), n=1..12); # Ridouane Oudra, Oct 04 2022

Mathematica

Flatten[Table[PowerMod[n,-1,p],{p,Prime[Range[9]]},{n,p-1}]] (* Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010 *)
T[n_, k_] := ModularInverse[k, Prime[n]]; Table[T[n, k], {n, 1, 9}, {k, 1, Prime[n]-1}] // Flatten (* Jean-François Alcover, May 08 2017 *)

PARI

row(n) = my(p=prime(n)); vector(p-1, k, lift(1/Mod(k, prime(n)))); \\ Michel Marcus, Feb 24 2023

Formula

From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)

T(n, 1) = 1;

T(n, T(n,k)) = k;

T(n, prime(n)-1) = prime(n)-1. (End)

T(n,k) = k^(prime(n)-2) mod prime(n), with 1 <= k < prime(n). - Ridouane Oudra, Oct 04 2022

From Ammar Khatab, Nov 07 2024: (Start)

T(n,2) = (prime(n)+1)/2;

T(n,3) = (2*prime(n)+1)/3 + 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);

T(n,8) = (8*prime(n)+1)/8 - prime(n)/8 * (prime(n) mod 8);

T(n,prime(n)-k) = prime(n) - T(n,k);

T(n,prime(n)-2) = (prime(n)-1)/2 ;

T(n,prime(n)-3) = (prime(n)-1)/3 - 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);

T(n,prime(n)-8) = -1/8 + prime(n)/8 * (prime(n) mod 8). (End)