A289251 Triangle T(n, k), n > 0 and 0 <= k < n, read by rows; if gcd(n, k) = 1, then T(n, k) = modular inverse of k (mod n), otherwise T(n, k) = k.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 2, 4, 0, 1, 2, 3, 4, 5, 0, 1, 4, 5, 2, 3, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 5, 3, 7, 2, 6, 4, 8, 0, 1, 2, 7, 4, 5, 6, 3, 8, 9, 0, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 7, 9, 10, 8, 11

Offset: 1

Author

Rémy Sigrist, Jun 29 2017

Keywords

• nonn
• tabl

Comments

The n-th row has n terms, and is a self-inverse permutation of the first n nonnegative numbers.

T(n, 0) = 0 for any n > 0.

T(n, 1) = 1 for any n > 1.

T(n, n-1) = n-1 for any n > 0.

If n > 0 and gcd(n, k) = 1 then T(n, k) = A102057(n, k).

T(prime(n), k) = A124223(n, k) for any n > 0 and k in 1..prime(n)-1.

Examples

			The first rows are:
n\k  0 1 2 3 4 5 6 7 8 9
1    0
2    0 1
3    0 1 2
4    0 1 2 3
5    0 1 3 2 4
6    0 1 2 3 4 5
7    0 1 4 5 2 3 6
8    0 1 2 3 4 5 6 7
9    0 1 5 3 7 2 6 4 8
10   0 1 2 7 4 5 6 3 8 9
		

Links

• Rémy Sigrist, Rows n=1..100 of triangle, flattened

Crossrefs

Cf. A102057, A124223.

Programs

Mathematica

T[n_, k_] := If[GCD[n, k] == 1, PowerMod[k, -1, n], k];
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 31 2017 *)

PARI

T(n, k) = if (gcd(n, k)==1, lift(1/Mod(k, n)), k)