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

1,3

Comments

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

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

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)

Example

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A003963 A003960 A243499 * A377027 A295908 A300980

Adjacent sequences: A124220 A124221 A124222 * A124224 A124225 A124226

Keyword

nonn,tabf,changed

Author

Franklin T. Adams-Watters, Oct 20 2006

Status

approved