A178151 The number of quadratic residues (mod p) less than p/2, where p=prime(n).

1, 1, 2, 4, 3, 4, 6, 7, 7, 9, 9, 10, 12, 14, 13, 19, 15, 18, 21, 18, 22, 25, 22, 24, 25, 28, 31, 27, 28, 34, 40, 34, 39, 37, 41, 39, 42, 47, 43, 52, 45, 54, 48, 49, 54, 57, 59, 64, 57, 58, 67, 60, 73, 64, 72, 67, 73, 69, 70, 75, 73, 81, 87, 78, 79, 87, 84, 94, 87, 88, 99, 96, 93

Offset: 2

Author

T. D. Noe, May 21 2010

Keywords

• nonn

Comments

Sequence A063987 lists the quadratic residues (mod p) for each prime p. When p=1 (mod 4), there are an equal number of quadratic residues less than p/2 and greater than p/2. When p=3 (mod 4), there are always more quadratic residues less than p/2 than greater than p/2.

Examples

			The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Six of these are less than 19/2. Hence a(8)=6.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 2..3132
• MathOverflow, Most squares in the first half-interval

Crossrefs

Cf. A178152, A178153, A178154

Programs

Maple

A178151 := proc(n)
    local r,a,p;
    p := ithprime(n) ;
    a := 0 ;
    for r from 1 to p/2 do
        if numtheory[legendre](r,p) =1 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Feb 10 2017

Mathematica

Table[p=Prime[n]; Length[Select[Range[(p-1)/2], JacobiSymbol[ #,p]==1&]], {n,2,100}]