A060504 -id:A060504 - cp's OEIS frontend

A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.

2, 40487, 6692367337

Offset: 1

Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000

For r a primitive root of a prime p, r + qp is a primitive root of p: but r + qp is also a primitive root of p^2, except for q in some unique residue class modulo p. In the exceptional case, r + qp has order p-1 modulo p^2 (Burton, section 8.3).
No other terms below 10^12 (Paszkiewicz, 2009).
Each term p is a Wieferich prime to base A046145(p). For example, a(2) and a(3) are base-5 Wieferich (see A123692). - Jeppe Stig Nielsen, Mar 06 2020

David Burton, Elementary Number Theory, Allyn and Bacon, Boston, 1976, first edition (cf. Section 8.3).

Joerg Arndt, Matters Computational (The Fxtbook), section 39.7.2, p.780.
Stephen Glasby, Three questions about the density of certain primes, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Apr 22, 2001.
Bryce Kerr, Kevin McGown, Tim Trudgian, The least primitive root modulo p^2, arXiv:1908.11497 [math.NT], 2019.
A. Paszkiewicz, A new prime for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.

Cf. A060503, A060504.

Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* Arkadiusz Wesolowski, Sep 06 2012 *)

Prime A000040(n) is in this sequence iff A001918(n)^(A000040(n)-1) == 1 (mod A000040(n)^2).

Prime A000040(n) is in this sequence iff A001918(n) differs from A127807(n).

a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001

Edited by Max Alekseyev, Nov 10 2011

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

Original entry on oeis.org

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

Jud McCranie, Mar 22 2001

nonn

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

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

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

Cf. A055578, A060504, A134307.

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

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 *)

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

A101710 Primes p for which the least-magnitude negative primitive root is not a primitive root of p^2. Like A055578, but for negative rather than positive primitive roots.

Original entry on oeis.org

3, 11, 3511, 6692367337

Offset: 1

Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Dec 13 2004

There is a rough heuristic suggesting that a prime p will occur in this list with probability 1/p; the actual density seen here tails off faster than that. No other primes with this property exist up to 2^36. Used for testing a multiprecision division algorithm.

The sequence giving the least-magnitude primitive roots r of primes p for which r is not a primitive root of p^2 begins -1,-3,-2,-5,..., with no other cases known up to 2^36.

			-3 is a primitive root of 11. That is, the successive powers of -3 work through all the nonzero residues modulo 11 before coming round through 1 to -3 again: -3, -2, -5, 4, -1, 3, 2, 5, -4, 1, -3, ...
-3 also happens to be the negative number of least magnitude with this property (-1 obviously fails, -2 yields -2, 4, 3, 5, 1, -2 ...) Modulo 11^2 = 121, however, successive powers of -3 do not yield all the corresponding residues (that is, all the ones which aren't multiples of 11): we only get -3, 9, -27, 81, -1, 3, -9, 27, -81, 1, -3, ...

Cf. A055578, A060503. A060504.

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A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.

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

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A101710 Primes p for which the least-magnitude negative primitive root is not a primitive root of p^2. Like A055578, but for negative rather than positive primitive roots.

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