A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n).

1, 0, 1, 3, 0, 0, 3, 3, 0, 3, 0, 0, 3, 5, 0, 9, 0, 3, 7, 0, 5, 9, 0, 0, 0, 5, 9, 0, 0, 5, 15, 0, 9, 0, 7, 0, 3, 11, 0, 15, 0, 13, 0, 0, 9, 9, 7, 15, 0, 0, 15, 0, 21, 0, 13, 0, 11, 0, 0, 9, 0, 9, 19, 0, 0, 9, 0, 15, 0, 0, 19, 9, 0, 9, 17, 0, 0, 0, 0, 27, 0, 21, 0, 15, 15, 0, 0, 0, 7, 21, 25, 7, 27, 9, 21, 0

Offset

2,4

Comments

When prime(n)=1 (mod 4), then a(n)=0. When prime(n)=3 (mod 4), then a(n)>0. When prime(n)=3 (mod 8) and prime(n)>3, then 3 divides a(n). See Borevich and Shafarevich. The nonzero terms of this sequence are closely related to A002143, the class number of primes p=3 (mod 4).

Same as difference between the numbers of quadratic residues and nonresidues (mod p) less than p/2, where p=prime(n). - Jonathan Sondow, Oct 30 2011

References

Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966, p. 346.

H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.

Links

R. J. Mathar, Table of n, a(n) for n = 2..2066

MathOverflow, Most squares in the first half-interval

Formula

a(n) = A178151(n) - A178152(n).

a(n) = sum(j=1..(p-1)/2, (j|p)), where p = prime(n) and (j|p) = +/-1 is the Legendre symbol. - Jonathan Sondow, Oct 30 2011

Example

The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Hence a(8)=6-3=3.

Maple

A178153 := 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:
    for r from ceil(p/2) to p-1 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&]] - Length[Select[Range[(p+1)/2, p-1], JacobiSymbol[ #, p]==1&]], {n, 2, 100}]
Table[p = Prime[n]; Sum[ JacobiSymbol[a, p], {a, 1, (p-1)/2}], {n, 2, 100}] (* Jonathan Sondow, Oct 30 2011 *)

Crossrefs

Cf. A178154 (without the zero terms).

Sequence in context: A230652 A230661 A178952 * A267794 A282499 A216194

Adjacent sequences: A178150 A178151 A178152 * A178154 A178155 A178156

Keyword

nonn

Author

T. D. Noe, May 21 2010

Status

approved