%I #36 Mar 29 2020 18:35:36

%S 2,40487,6692367337

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

%C 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).

%C No other terms below 10^12 (Paszkiewicz, 2009).

%C 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

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

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.7.2, p.780.

%H Stephen Glasby, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;e117c4a.0104">Three questions about the density of certain primes</a>, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Apr 22, 2001.

%H Bryce Kerr, Kevin McGown, Tim Trudgian, <a href="https://arxiv.org/abs/1908.11497">The least primitive root modulo p^2</a>, arXiv:1908.11497 [math.NT], 2019.

%H A. Paszkiewicz, <a href="https://doi.org/10.1090/S0025-5718-08-02090-5">A new prime for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal</a>, Math. Comp. 78 (2009), 1193-1195.

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

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

%t Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* _Arkadiusz Wesolowski_, Sep 06 2012 *)

%Y Cf. A060503, A060504.

%K hard,nonn,bref,more

%O 1,1

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

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

%E Edited by _Max Alekseyev_, Nov 10 2011