cp's OEIS Frontend

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

%I A055578 #36 Mar 29 2020 18:35:36
%S A055578 2,40487,6692367337
%N A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.
%C A055578 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 A055578 No other terms below 10^12 (Paszkiewicz, 2009).
%C A055578 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 A055578 David Burton, Elementary Number Theory, Allyn and Bacon, Boston, 1976, first edition (cf. Section 8.3).
%H A055578 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.7.2, p.780.
%H A055578 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 A055578 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 A055578 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 A055578 Prime A000040(n) is in this sequence iff A001918(n)^(A000040(n)-1) == 1 (mod A000040(n)^2).
%F A055578 Prime A000040(n) is in this sequence iff A001918(n) differs from A127807(n).
%t A055578 Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* _Arkadiusz Wesolowski_, Sep 06 2012 *)
%Y A055578 Cf. A060503, A060504.
%K A055578 hard,nonn,bref,more
%O A055578 1,1
%A A055578 Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000
%E A055578 a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001
%E A055578 Edited by _Max Alekseyev_, Nov 10 2011