A108989 Composite numbers k with primitive root 2; i.e., the order of 2 modulo k is phi(k).

9, 25, 27, 81, 121, 125, 169, 243, 361, 625, 729, 841, 1331, 1369, 2187, 2197, 2809, 3125, 3481, 3721, 4489, 6561, 6859, 6889, 10201, 11449, 14641, 15625, 17161, 19321, 19683, 22201, 24389, 26569, 28561, 29929, 32041, 32761, 38809, 44521, 50653

Offset: 1

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Jul 28 2005

Keywords

• nonn

Comments

There exist no even numbers with primitive root 2. All entries are odd. They are all the powers of odd primes. - V. Raman, Nov 20 2012

Examples

			Modulo 9: 2^1 == 2, 2^2 == 4, 2^3 == 8, 2^4 == 7, 2^5 == 5, 2^6 == 1 and phi(9) == 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Alois P. Heinz)

Crossrefs

Intersection of A002808 and A167791.

Cf. A000010, A001122, A002326, A216838, A216848, A219030.

Programs

GAP

for i in [2..100000] do if not IsPrime(i) then if IsPrimitiveRootMod(2,i) then Display(i); fi; fi; od;

Mathematica

nn=51000; Select[Complement[Range[2, nn], Prime[Range[PrimePi[nn]]]], PrimitiveRoot[#] == 2&] (* Harvey P. Dale, Jul 25 2011 *)
seq[max_] := Module[{ps = Select[Range[2, Floor[Sqrt[max]]], PrimeQ], s = {}}, Do[s = Join[s, Select[p^Range[2, Floor[Log[p, max]]], PrimitiveRoot[#] == 2 &]], {p, ps}]; Sort[s]]; seq[10^5] (* Amiram Eldar, Nov 10 2023 *)

PARI

for(n=3,100000,if(n%2==1&&isprime(n)==0&&znorder(Mod(2,n))==eulerphi(n),print1(n","))) /* V. Raman, Nov 20 2012 */