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A275111 a(n) = prime(n)! mod prime(n+1).

%I A275111 #30 Feb 24 2023 02:31:10
%S A275111 2,1,1,2,1,3,1,4,22,1,33,7,1,8,19,30,1,43,12,1,27,14,23,24,17,1,18,1,
%T A275111 19,19,22,8,1,94,1,140,72,28,62,91,1,105,1,33,1,177,97,38,1,39,2,1,19,
%U A275111 15,160,204,1,247,47,1,291,299,52,1,53,198,132,55,1,59,3,176
%N A275111 a(n) = prime(n)! mod prime(n+1).
%C A275111 By Wilson's theorem, if prime(n+1) - prime(n) = 2 then a(n) = 1.
%C A275111 However a(991) = 1, while prime(992) - prime(991) = 7853 - 7841 = 12. See A286181, A286208, A286230. - _Robert Israel_, Jul 17 2016
%H A275111 Chai Wah Wu, <a href="/A275111/b275111.txt">Table of n, a(n) for n = 1..10000</a>
%F A275111 For n>1, a(n) = 1/((prime(n)+1)*(prime(n)+2)*...*(prime(n+1)-2)) mod prime(n+1). - _Robert Israel_, Jul 17 2016; corrected by _Max Alekseyev_, May 03 2017
%F A275111 For n>1, a(n) = 1/(prime(n+1)-prime(n)-1)! mod prime(n+1) = 1/(A001223(n)-1)! mod A000040(n+1). - _Max Alekseyev_, May 03 2017
%t A275111 Table[Mod[#!, NextPrime@ #] &@ Prime@ n, {n, 120}] (* _Michael De Vlieger_, Jul 17 2016 *)
%o A275111 (PARI) a(n) = prime(n)! % prime(n+1); \\ _Michel Marcus_, Jul 17 2016
%o A275111 (PARI) a(n,p=prime(n))=my(q=nextprime(p+1)); if(p==2, 2, lift( 1/prod(r=p+1,q-2, Mod(r,q)) ) ); \\ _Charles R Greathouse IV_, Jul 18 2016; corrected by _Max Alekseyev_, May 03 2017
%o A275111 (PARI) a(n,p=prime(n)) = my(q=nextprime(p+1)); if(p==2, 2, (1/(q-p-1)!)%q); \\ _Max Alekseyev_, May 03 2017
%o A275111 (Python)
%o A275111 from sympy import prime
%o A275111 from sympy.core.numbers import igcdex
%o A275111 def A275111(n):
%o A275111     p, q = prime(n), prime(n+1)
%o A275111     a = q-1
%o A275111     for i in range(p+1,q):
%o A275111         a = (a*igcdex(i,q)[0]) % q
%o A275111     return a # _Chai Wah Wu_, Jul 18 2016
%o A275111 (Python)
%o A275111 from functools import reduce
%o A275111 from sympy import prime
%o A275111 def A275111(n): return ((q:=prime(n+1))-1)*pow(reduce(lambda i,j:i*j%q,range(prime(n)+1,q),1),-1,q)%q # _Chai Wah Wu_, Feb 24 2023
%Y A275111 Cf. A286181, A286208, A286230.
%K A275111 nonn
%O A275111 1,1
%A A275111 _Thomas Ordowski_, Jul 17 2016
%E A275111 More terms from _Altug Alkan_, Jul 17 2016