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 A212158 #16 Jun 13 2025 01:09:18 %S A212158 1,2,6,120,720,40320,362880,39916800,87178291200,1307674368000, %T A212158 6402373705728000,2432902008176640000,51090942171709440000, %U A212158 25852016738884976640000,403291461126605635584000000,8841761993739701954543616000000 %N A212158 a(n) = ((prime(n) - 1)/2)!, n >= 2. %C A212158 a(n)^2 == (-1)^((prime(n) + 1)/2) (mod prime(n)). %C A212158 Use product(p-j,j=1..(p-1)/2) == (-1)^((p-1)/2)*a(n) (mod p) for p=prime(n), n>=2, hence a(n)*(-1)^((p-1)/2)*a(n) == (p-1)! (mod p), then apply Wilson's theorem. That is, a(n)^2 == -1 (mod prime(n)) for primes of the form 4*k+1 (see A002144) and +1 for primes of the form 4*k+3 (see A002145). See the link with a blog by W. Holsztyński. %C A212158 See A004055 for a(n) (mod prime(n)), n>=2. %C A212158 See A212159 for a(n)^2 (mod prime(n)), n>=2. %H A212158 Holsztyński Włodzimierz, <a href="http://wlod.wordpress.com/article/congruence-x-2-1-mod-p-euler-and-a-1jxfhq4x4sw0j-65/">Congruence x^2==-1 (mod p) (Euler), and a super-Wilson Theorem</a> %F A212158 a(n) = ((prime(n) - 1)/2)!, n>=2, with prime(n) = A000040(n). %F A212158 a(n) = A005097(n-1)!, n>=2. %e A212158 a(4) = ((7-1)/2)! = 3! = 6. %e A212158 a(4)^2 = 36 == +1 (mod 7), because (7 + 1)/2 = 4, and 4 is even. %e A212158 a(6) = ((13-1)/2)! = 6! = 720. %e A212158 a(6)^2 = 518400 == -1 (mod 13) = 12 (mod 13) because (13+1)/2 = 7, and 7 is odd. %t A212158 ((Prime[Range[2,20]]-1)/2)! (* _Harvey P. Dale_, Jan 24 2021 *) %K A212158 nonn,easy %O A212158 2,2 %A A212158 _Wolfdieter Lang_, May 08 2012