A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

0, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1

Offset: 1

Author

Reinhard Zumkeller, May 04 2002

Keywords

• sign
• nice
• easy

Comments

Also, sin(prime(n)*Pi/2), where prime(n) = A000040(n), Pi=3.1415... (original definition).

Also imaginary part of primes mapped as defined in A076340, A076341: a(n) = A076341(A000040(n)), real part = A076342.

Legendre symbol (-1/prime(n)) for n > 1. - T. D. Noe, Nov 05 2003

For n > 1, let p = prime(n) and m = (p-1)/2. Then c(m) - a(n) == 0 (mod p), where c(m) = (2*m)!/(m!)^2 = A000984(m) is the central binomial coefficient. [Proof: By definition, c(m)*(m!)^2 - (p-1)! = 0 and therefore c(m)*(m!)^2*(-1)^(m+1) - (p-1)!*(-1)^(m+1) = 0. Now apply Wilson's theorem, (p-1)! == 1 (mod p), and its corollary, (m!)^2 == (-1)^(m+1) (mod p), and finally use the formula by T. D. Noe listed below to replace (-1)^m with a(n).] Similarly, C_m - 2*a(n) == 0 (mod p), with C_m = A000108(m) being the m-th Catalan number. [Proof: By definition, C_m*(p+1)*(m!)^2 - 2*(p-1)! = 0. The result follows proceeding as in the first proof.] - Stanislav Sykora, Aug 11 2014

Examples

			p = 4*k+1 (see A002144): a(p) = sin((4*k+1)*Pi/2) = sin(2*k*Pi + Pi/2) = sin(Pi/2) = 1.
p = 4*k+3 (see A002145): a(p) = sin((4*k+3)*Pi/2) = sin(2*k*Pi + 3*Pi/2) = sin(3*Pi/2) = -1.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Legendre Symbol.
• Wikipedia, Wilson's theorem.

Crossrefs

Cf. A000040, A039702, A070748, A070749, A002144, A002145, A000108, A000984, A134323, A257834, A076340, A076341.

Cf. A060294, A088538.

Programs

Haskell

a070750 = (2 -) . (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 28 2012

Mathematica

a[n_] := JacobiSymbol[-1, Prime[n]]; a[1] = 0; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 05 2012, after T. D. Noe *)
Table[Which[EvenQ[p],0,Mod[p,4]==1,1,True,-1],{p,Prime[Range[80]]}] (* Harvey P. Dale, Mar 16 2020 *)

PARI

apply(n->2-n%4,primes(100)) \\ Charles R Greathouse IV, Aug 21 2011

Formula

a(n) = 2 - prime(n) mod 4 = 2 - A039702(n).

a(n) = (-1)^((prime(n)-1)/2) for n > 1. - T. D. Noe, Nov 05 2003

From Amiram Eldar, Dec 24 2022: (Start)

Product_{n>=1} (1 - a(n)/prime(n)) = 4/Pi (A088538).

Product_{n>=1} (1 + a(n)/prime(n)) = 2/Pi (A060294). (End)

Extensions

Wording of definition changed by N. J. A. Sloane, Jun 21 2015