A125616 (Sum of the quadratic nonresidues of prime(n)) / prime(n).

1, 2, 3, 3, 4, 5, 7, 7, 9, 9, 10, 11, 14, 13, 16, 15, 17, 21, 18, 22, 22, 22, 24, 25, 28, 28, 27, 28, 34, 35, 34, 36, 37, 41, 39, 41, 47, 43, 47, 45, 54, 48, 49, 54, 54, 59, 59, 57, 58, 67, 60, 66, 64, 72, 67, 73, 69, 70, 72, 73, 78, 87, 78, 79, 84, 84, 89, 87, 88, 99, 96, 93, 96

Offset: 3

Author

Nick Hobson, Nov 30 2006

Keywords

• easy
• nonn

Comments

Always an integer for primes >= 5.

Examples

			The quadratic nonresidues of 7=prime(4) are 3, 5 and 6. Hence a(4) = (3+5+6)/7 = 2.
		

References

• D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 185.

Links

• Nick Hobson, Table of n, a(n) for n = 3..1000

Crossrefs

Cf. A002143, A076409, A076410, A125613-A125618.

Programs

Maple

a:= proc(n) local p;
   p:= ithprime(n);
   convert(select(t->numtheory:-legendre(t,p)=-1, [$1..p-1]),`+`)/p;
end proc:
seq(a(n),n=3..100); # Robert Israel, May 10 2015

Mathematica

Table[Total[Flatten[Position[Table[JacobiSymbol[a, p], {a, p - 1}], -1]]]/ p, {p, Prime[Range[3, 100]]}] (* Geoffrey Critzer, May 10 2015 *)

PARI

vector(73, m, p=prime(m+2); t=1; for(i=2, (p-1)/2, t+=((i^2)%p)); (p-1)/2-t/p)

Formula

a(n) = A125615(n)/prime(n).

If prime(n) = 4k+1 then a(n) = k = A076410(n).