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A267133 a(n) = (1/n)(2/n)(3/n)...((n-1)/n) where (k/n) is the Kronecker symbol, n >= 1.

%I A267133 #46 Jan 02 2019 11:40:43
%S A267133 1,1,-1,0,1,0,-1,0,0,0,-1,0,1,0,0,0,1,0,-1,0,0,0,-1,0,0,0,0,0,1,0,-1,
%T A267133 0,0,0,0,0,1,0,0,0,1,0,-1,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,
%U A267133 0,0,-1,0,0,0,-1,0,1,0,0
%N A267133 a(n) = (1/n)(2/n)(3/n)...((n-1)/n) where (k/n) is the Kronecker symbol, n >= 1.
%H A267133 Robert Israel, <a href="/A267133/b267133.txt">Table of n, a(n) for n = 1..10000</a>
%F A267133 A080339(n) = abs(a(n)) = a(n)^2.
%F A267133 a(c) = 0 if c is composite (A002808).
%F A267133 a(p) = 1 for primes p in A002313.
%F A267133 a(p) = -1 for primes p in A002145.
%F A267133 a(n) = A057077(n+3)*A080339(n) for n > 1. - _Robert Israel_, Jan 14 2016
%F A267133 a(n) = A151763(n), n > 2. - _R. J. Mathar_, Jan 17 2016
%e A267133 a(3) = (1/3)(2/3) = (1)(-1) = -1.
%p A267133 f:= proc(n) if not isprime(n) then 0 elif n mod 4 = 3 then -1 else 1 fi end proc:
%p A267133 f(1):= 1:
%p A267133 map(f, [$1..1000]); # _Robert Israel_, Jan 14 2016
%t A267133 Table[Product[JacobiSymbol[k, n], {k, n - 1}], {n, 75}] (* _Michael De Vlieger_, Jan 12 2016 *)
%o A267133 (PARI) a(n) = prod(k=1, n-1, kronecker(k, n)); \\ _Michel Marcus_, Jan 11 2016
%o A267133 (PARI) a(n)=if(isprime(n),(-1)^(n%4>2),n==1) \\ _Charles R Greathouse IV_, Jan 14 2016
%Y A267133 Cf. A002313, A002145, A002808, A057077, A080339, A151763.
%K A267133 sign,easy
%O A267133 1,1
%A A267133 _Dimitri Papadopoulos_, Jan 10 2016
%E A267133 "Jacobi symbol" in Name changed to "Kronecker symbol" by _Jianing Song_, Dec 30 2018