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A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2. 8
2, 40487, 6692367337 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..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.

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.

FORMULA

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).

MATHEMATICA

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

CROSSREFS

Cf. A060503, A060504.

Sequence in context: A206854 A272166 A257968 * A232733 A106025 A157959

Adjacent sequences:  A055575 A055576 A055577 * A055579 A055580 A055581

KEYWORD

hard,nonn,bref,more

AUTHOR

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

EXTENSIONS

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

Edited by Max Alekseyev, Nov 10 2011

STATUS

approved

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Last modified December 4 21:33 EST 2016. Contains 278755 sequences.