A124224 Table T(n,k) = reciprocal of k-th number prime to n, modulo n, for 1 <= k <= phi(n).

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, 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, 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

Offset: 1

Author

Franklin T. Adams-Watters, Oct 20 2006

Keywords

• nonn
• tabf

Comments

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

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

Examples

			The table T(n,k) starts:
n\k 1  2  2  3 4  5 6  7 8  9 10 11
1:  0
2:  1
3:  1  2
4:  1  3
5:  1  3  2  4
6:  1  5
7:  1  4  5  2 3  6
8:  1  3  5  7
9:  1  5  7  2 4  8
10: 1  7  3  9
11: 1  6  4  3 9  2 8  7 5 10
12: 1  5  7 11
13: 1  7  9 10 8 11 2  5 3  4  6 12
14: 1  5  3 11 9 13
15: 1  8  4 13 2 11 7 14
16: 1 11 13  7 9  3 5 15
...
n = 17: 1  9  6 13 7  3  5 15 2 12 14 10 4 11 8 16,
n = 18: 1 11 13  5 7 17,
n = 19: 1 10 13  5 4 16 11 12 17 2 7 8 3 15 14 6 9 18,
n = 20: 1 7 3 9 11 17 13 19.
... reformatted (extended and corrected), - _Wolfdieter Lang_, Oct 06 2016
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10060
• Eric Weisstein's World of Mathematics, Modular Inverse

Crossrefs

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

Programs

Maple

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

Mathematica

Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
PowerMod[a, -1, nn], {n, 1, 20}] // Grid (* Geoffrey Critzer, Jan 03 2015 *)

Formula

T(n,k) * A038566(n,k) = 1 (mod n), for n >=1 and k=1..A000010(n). - Wolfdieter Lang, Oct 06 2016