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A055578
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"Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.
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8
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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David Burton, Elementary Number Theory, Allyn and Bacon, Boston, 1976, first edition (cf. Section 8.3).
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LINKS
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Table of n, a(n) for n=1..3.
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.
Joerg Arndt, Fxtbook, section 39.7.2, p.780.
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FORMULA
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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).
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MATHEMATICA
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Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* Arkadiusz Wesolowski, Sep 06 2012 *)
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CROSSREFS
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Cf. A060503, A060504.
Sequence in context: A030462 A001377 A206854 * A106025 A157959 A094213
Adjacent sequences: A055575 A055576 A055577 * A055579 A055580 A055581
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KEYWORD
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hard,nonn,bref,more
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AUTHOR
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Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000
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EXTENSIONS
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a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001
Edited by Max Alekseyev, Nov 10 2011
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STATUS
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approved
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