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

%I A124223 #54 Feb 16 2025 08:33:03
%S A124223 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,
%T A124223 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,
%U A124223 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
%N A124223 Table T(n,k) = reciprocal of k modulo prime(n), for 1 <= k < prime(n), read by rows.
%C A124223 T(n,k) = smallest m such that k*m == 1 (mod prime(n)); prime(n) is the n-th prime: A000040(n).
%H A124223 Franklin T. Adams-Watters, <a href="/A124223/b124223.txt">Table of n, a(n) for n = 1..4181 (primes less than 200)</a>
%H A124223 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModularInverse.html">Modular Inverse</a>
%F A124223 From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)
%F A124223 T(n, 1) = 1;
%F A124223 T(n, T(n,k)) = k;
%F A124223 T(n, prime(n)-1) = prime(n)-1. (End)
%F A124223 T(n,k) = k^(prime(n)-2) mod prime(n), with 1 <= k < prime(n). - _Ridouane Oudra_, Oct 04 2022
%F A124223 From _Ammar Khatab_, Nov 07 2024: (Start)
%F A124223 T(n,2) = (prime(n)+1)/2;
%F A124223 T(n,3) = (2*prime(n)+1)/3 + 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);
%F A124223 T(n,8) = (8*prime(n)+1)/8 - prime(n)/8 * (prime(n) mod 8);
%F A124223 T(n,prime(n)-k) = prime(n) - T(n,k);
%F A124223 T(n,prime(n)-2) = (prime(n)-1)/2 ;
%F A124223 T(n,prime(n)-3) = (prime(n)-1)/3 - 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);
%F A124223 T(n,prime(n)-8) = -1/8 + prime(n)/8 * (prime(n) mod 8). (End)
%e A124223 From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)
%e A124223 Table begins:
%e A124223   1;
%e A124223   1,2;
%e A124223   1,3,2,4;
%e A124223   1,4,5,2,3,6;
%e A124223   1,6,4,3,9,2,8,7,5,10;
%e A124223   1,7,9,10,8,11,2,5,3,4,6,12;
%e A124223   1,9,6,13,7,3,5,15,2,12,14,10,4,11,8,16;
%e A124223   1,10,13,5,4,16,11,12,17,2,7,8,3,15,14,6,9,18;
%e A124223   1,12,8,6,14,4,10,3,18,7,21,2,16,5,20,13,19,9,17,15,11,22;
%e A124223   ... (End)
%p A124223 seq(seq(k^(ithprime(n)-2) mod ithprime(n), k=1..ithprime(n)-1), n=1..12); # _Ridouane Oudra_, Oct 04 2022
%t A124223 Flatten[Table[PowerMod[n,-1,p],{p,Prime[Range[9]]},{n,p-1}]] (* Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010 *)
%t A124223 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 *)
%o A124223 (PARI) row(n) = my(p=prime(n)); vector(p-1, k, lift(1/Mod(k, prime(n)))); \\ _Michel Marcus_, Feb 24 2023
%Y A124223 Cf. A124224, A102057, A000040, A006093 (row lengths).
%K A124223 nonn,tabf
%O A124223 1,3
%A A124223 _Franklin T. Adams-Watters_, Oct 20 2006