A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

1, 2, 2, 3, 3, 5, 2, 8, 7, 6, 2, 6, 11, 7, 3, 10, 5, 11, 14, 7, 12, 6, 2, 13, 14, 15, 3, 10, 5, 10, 20, 17, 11, 21, 19, 15, 7, 14, 2, 8, 3, 19, 18, 14, 27, 21, 26, 10, 11, 15, 3, 17, 13, 24, 22, 12, 11, 21, 2, 32, 17, 13, 15, 18, 35, 5, 20, 24, 22, 19

Offset: 1

Author

Geoffrey Critzer, May 03 2015

Keywords

• nonn
• tabf

Comments

Each row is a complete set of incongruent primitive roots.

Each row is a permutation of the corresponding row in A060749.

Row lengths are A008330.

T(n,1) = A001918(n).

Examples

			1;
2;
2,  3;
3,  5;
2,  8,  7,  6;
2,  6, 11,  7;
3, 10,  5, 11, 14,  7, 12,  6;
2, 13, 14, 15,  3, 10;
5, 10, 20, 17, 11, 21, 19, 15,  7, 14;
2,  8,  3, 19, 18, 14, 27, 21, 26, 10, 11, 15;
Row 6 contains 2,6,11,7 because 13 is the 6th prime number. 2 is the least positive primitive root of 13. The integers relatively prime to 13-1=12 are {1,5,7,11}. So we have: 2^1==2, 2^5==6, 2^7==11, and 2^11==7 (mod 13).
		

Links

• Alois P. Heinz, Rows n = 1..120, flattened

Crossrefs

Cf. A001918, A008330, A060749.

Last elements of rows give A255367.

Row sums give A088144.

Programs

Maple

with(numtheory):
T:= n-> (p-> seq(primroot(p)&^k mod p, k=select(
         h-> igcd(h, p-1)=1, [$1..p-1])))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, May 03 2015

Mathematica

Table[nn = p;Table[Mod[PrimitiveRoot[nn]^k, nn], {k,Select[Range[nn - 1], CoprimeQ[#, nn - 1] &]}], {p,Prime[Range[12]]}] // Grid