A179858 Least positive primitive root of A139035(n).

3, 5, 5, 7, 3, 5, 5, 19, 3, 7, 5, 6, 17, 7, 6, 5, 3, 13, 3, 5, 7, 3, 5, 11, 5, 3, 3, 11, 5, 5, 5, 5, 6, 14, 3, 3, 3, 17, 5, 3, 3, 6, 13, 5, 7, 3, 5, 11, 5, 19, 3, 5, 5, 3, 6, 10, 5, 5, 14, 6, 3, 7, 5, 5, 7, 5, 3, 3, 11, 5, 5, 3, 5, 6, 7, 3, 5, 7, 3, 7, 5, 5, 5, 17

Offset: 1

Author

Vladimir Shevelev, Jan 11 2011

Keywords

• nonn

Comments

If p is a prime, then a is called a semiprimitive root if it has order (p-1)/2 and there is no x for which a^x is congruent to -1 (mod p). So +- a^k, 0 <= k <= (p-3)/2 is a complete set of nonzero residues (mod p). A primitive root has order p-1, so a number cannot be both a primitive root and a semiprimitive root.

A139035 are the primes for which 2 is a semiprimitive root. This sequence gives the smallest positive primitive root corresponding to each term of A139035, so each term is greater than or equal to 3.

Examples

			Since A139035(13)=311, 2 is a semiprimitive root of 311 so j=0,...,154, {+-2^j} is a complete set of residues (congruent to {1,...,310}).  The corresponding member of this sequence is a(13)=17 because 17 is the smallest positive integer a for which {a^k}, k=0,...,309 is a complete set of residues.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001918, A139035.

Programs

Mathematica

PrimitiveRoot /@ Reap[For[p = 3, p < 3000, p = NextPrime[p], rp = MultiplicativeOrder[2, p]; rm = MultiplicativeOrder[-2, p]; If[rp != p-1 && rm == p-1, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Sep 03 2016, after Joerg Arndt's code for A139035 *)

Extensions

More terms from Jean-François Alcover, Sep 03 2016