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 A228120 #26 May 16 2025 00:06:29
%S A228120 2,10,100,44200,1635400,5315050000,435834100000,5370347780200000,
%T A228120 26078677338040210000000,5893781078397087460000000,
%U A228120 142760638726203851727985000000000,20723419838773203524537758570000000000,9159751568737755957845689287940000000000,2354514140744040168964234431464977000000000000
%N A228120 a(n) = (1^2 + 1)*(2^2 + 1)*(3^2 + 1)*...*(((prime(n) - 1)/2)^2 + 1).
%C A228120 The author has shown that a(n) == 2 (mod p_n) if p_n == 3 (mod 4). He has also established the following general theorem:
%C A228120 Let p be any odd prime and let d be any quadratic non-residue modulo p. Then we have the congruence
%C A228120 Product_{x=1..(p-1)/2} (x^2 - d) == 2*(-1)^((p+1)/2) (mod p).
%C A228120 This can be proved as follows: By Wilson's theorem we have (((p-1)/2)!)^2 == (-1)^((p+1)/2) (mod p), and thus we reduce the desired congruence to
%C A228120 Product_{0<k<p,(k/p)=-1} (1 - k) == 2 (mod p). (*)
%C A228120 Clearly
%C A228120 Product_{1<k<p, (k/p)=1} (1 - k)
%C A228120 == Product_{j=2..(p-1)/2} (1 - j^2)
%C A228120 = (-1)^((p+1)/2)*(((p-1)/2)!)^2*(p+1)/(2*p-2)
%C A228120 == -1/2 (mod p),
%C A228120 and Product_{k=2..p-1} (1 - k) = (-1)^(p-2)*(p-2)! == -1 (mod p) by Wilson's theorem. Therefore (*) follows.
%H A228120 Zhi-Wei Sun, <a href="/A228120/b228120.txt">Table of n, a(n) for n = 2..30</a>
%e A228120 a(3) = (1^1+1)*(2^2+1)*(3^2+1) = 100 and a(3) == 2 (mod 7).
%t A228120 a[n_]:=Product[(x^2+1),{x,1,(Prime[n]-1)/2}]
%t A228120 Table[a[n],{n,2,15}]
%Y A228120 Cf. A101686, A000040.
%K A228120 nonn
%O A228120 2,1
%A A228120 _Zhi-Wei Sun_, Aug 11 2013