This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079853 #44 Feb 23 2024 07:25:52
%S A079853 2,3,11,107,4931
%N A079853 Primes p for which (p-2)! == 1 (mod p^2).
%C A079853 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
%C A079853 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).
%C A079853 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)
%C A079853 No other terms below 4*10^11.
%C A079853 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
%H A079853 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wilson_prime#Near-Wilson_primes">Near-Wilson primes</a>
%t A079853 Select[Prime[Range[700]],Mod[(#-2)!,#^2]==1&] (* _Harvey P. Dale_, Jun 01 2014 *)
%o A079853 (PARI) forprime(n=2, 10^9, if(Mod((n-2)!, n^2)==1, print1(n, ", "))) \\ _Felix Fröhlich_, Jun 17 2014
%Y A079853 Cf. A002068, A007619, A128666.
%K A079853 nonn,more
%O A079853 1,1
%A A079853 Pavlos Saridis (pavlos19(AT)yahoo.com), Sep 13 2003
%E A079853 Edited by _Max Alekseyev_, Jan 28 2012