A060503 Primes p that have a primitive root between 0 and p that is not a primitive root of p^2.

2, 29, 37, 43, 71, 103, 109, 113, 131, 181, 191, 211, 257, 263, 269, 283, 349, 353, 359, 367, 373, 397, 439, 449, 461, 487, 509, 563, 599, 617, 619, 631, 641, 647, 653, 701, 739, 743, 773, 797, 839, 857, 863, 883, 887, 907, 919, 947, 971, 983, 1019, 1031

Offset

1,1

Comments

The smallest primitive roots of p that are not primitive roots of p^2 are in A060504.

Except for the initial term 2, this is a subsequence of A134307. - Jeppe Stig Nielsen, Jul 31 2015

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

14 is a primitive root of 29 but not of 29^2, so 29 is a term.

Maple

filter:= proc(p) local x;
  if not isprime(p) then return false fi;
  x:= 0;
  do
    x:= numtheory:-primroot(x, p);
    if x = FAIL then return false fi;
    if x &^ (p-1) mod p^2 = 1 then return true fi;
  od
end proc:
select(filter, [2, seq(i, i=3..2000, 2)]); # Robert Israel, Dec 01 2016

Mathematica

Reap[For[p = 2, p < 1100, p = NextPrime[p], prp = PrimitiveRootList[p]; prp2 = Select[PrimitiveRootList[p^2], # <= Last[prp]&]; If[AnyTrue[prp, FreeQ[prp2, #]&], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)

Prog

(PARI) forprime(p=2, , for(a=1, p-1, if(znorder(Mod(a, p))==p-1&Mod(a, p^2)^(p-1)==1, print1(p, ", "); break()))) \\ Jeppe Stig Nielsen, Jul 31 2015

Crossrefs

Cf. A055578, A060504, A134307.

Sequence in context: A041969 A295386 A019392 * A180231 A141172 A285688

Adjacent sequences: A060500 A060501 A060502 * A060504 A060505 A060506

Keyword

nonn

Author

Jud McCranie, Mar 22 2001

Status

approved