A124224 - OEIS

%I #22 Oct 19 2016 07:55:57

%S 0,1,1,2,1,3,1,3,2,4,1,5,1,4,5,2,3,6,1,3,5,7,1,5,7,2,4,8,1,7,3,9,1,6,

%T 4,3,9,2,8,7,5,10,1,5,7,11,1,7,9,10,8,11,2,5,3,4,6,12,1,5,3,11,9,13,1,

%U 8,4,13,2,11,7,14,1,11,13,7,9,3,5,15,1,9,6,13,7,3,5,15,2,12,14,10,4,11,8,16

%N Table T(n,k) = reciprocal of k-th number prime to n, modulo n, for 1 <= k <= phi(n).

%C T(n,k) = smallest m such that A038566(n,k) * m = 1 (mod n).

%C For n>1 every row begins with 1 and ends with n-1.  T(n,k) = A038566(n,k)^(phi(n) - 1) (mod n). - _Geoffrey Critzer_, Jan 03 2015

%H Robert Israel, <a href="/A124224/b124224.txt">Table of n, a(n) for n = 1..10060</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModularInverse.html">Modular Inverse</a>

%F T(n,k) * A038566(n,k) = 1 (mod n), for n >=1 and k=1..A000010(n). - _Wolfdieter Lang_, Oct 06 2016

%e The table T(n,k) starts:

%e n\k 1  2  2  3 4  5 6  7 8  9 10 11

%e 1:  0

%e 2:  1

%e 3:  1  2

%e 4:  1  3

%e 5:  1  3  2  4

%e 6:  1  5

%e 7:  1  4  5  2 3  6

%e 8:  1  3  5  7

%e 9:  1  5  7  2 4  8

%e 10: 1  7  3  9

%e 11: 1  6  4  3 9  2 8  7 5 10

%e 12: 1  5  7 11

%e 13: 1  7  9 10 8 11 2  5 3  4  6 12

%e 14: 1  5  3 11 9 13

%e 15: 1  8  4 13 2 11 7 14

%e 16: 1 11 13  7 9  3 5 15

%e ...

%e n = 17: 1  9  6 13 7  3  5 15 2 12 14 10 4 11 8 16,

%e n = 18: 1 11 13  5 7 17,

%e n = 19: 1 10 13  5 4 16 11 12 17 2 7 8 3 15 14 6 9 18,

%e n = 20: 1 7 3 9 11 17 13 19.

%e ... reformatted (extended and corrected), - _Wolfdieter Lang_, Oct 06 2016

%p 0,seq(seq(i^(-1) mod m, i = select(t->igcd(t,m)=1, [$1..m-1])),m=1..100); # _Robert Israel_, May 18 2014

%t Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];

%t PowerMod[a, -1, nn], {n, 1, 20}] // Grid (* _Geoffrey Critzer_, Jan 03 2015 *)

%Y Cf. A124223, A102057, A038566, A000010 (row lengths), A023896 (row sums after first)

%K nonn,tabf

%O 1,4

%A _Franklin T. Adams-Watters_, Oct 20 2006