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A079853
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Primes p for which (p-2)! == 1 (mod p^2).
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5
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OFFSET
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1,1
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COMMENTS
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These are generalized Wilson primes of order 2. Similarly to Wilson's theorem which states that (p-1)! == -1 (mod p) for every prime p>=n, we can prove that (n-1)!(p-n)! == (-1)^n (mod p) for every prime p. Generalized Wilson primes p of order n satisfy the recurrence (n-1)!(p-n)! == (-1)^n (mod p^2). Cf. A128666
Also, near-Wilson primes with Wilson quotient modulo p equals 1: prime p=prime(n) is in this sequence iff A002068(n) == A007619(n) == 1 (mod p).
Zhi-Wei SUN conjectures that for n>1, a(n) == 3 (mod 8). (Posting to the Number Theory Mailing List, Nov 02 2009; added by N. J. A. Sloane, Nov 02 2009)
No other terms below 4*10^11.
Conjecture: primes p such that Sum_{k=1..p-1} k^(1-p) == -1 (mod p^2) are the odd terms of this sequence. - Thomas Ordowski, Jul 02 2020
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LINKS
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MATHEMATICA
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Select[Prime[Range[700]], Mod[(#-2)!, #^2]==1&] (* Harvey P. Dale, Jun 01 2014 *)
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PROG
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(PARI) forprime(n=2, 10^9, if(Mod((n-2)!, n^2)==1, print1(n, ", "))) \\ Felix Fröhlich, Jun 17 2014
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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Pavlos Saridis (pavlos19(AT)yahoo.com), Sep 13 2003
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EXTENSIONS
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STATUS
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approved
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