A186708 Number of quadratic residues (mod p) in the interval [1,2k+1], for primes p=4k+3.

1, 2, 4, 6, 7, 9, 12, 14, 19, 18, 21, 22, 25, 28, 31, 34, 40, 39, 41, 42, 47, 52, 54, 54, 57, 59, 64, 67, 73, 72, 73, 75, 81, 87, 87, 94, 99, 96, 99, 104, 118, 118, 117, 118, 119, 127, 132, 125, 136, 129, 136, 138, 141, 154, 150, 157, 162

Offset

1,2

Comments

For primes of the form p=4k+3 (A002145), count numbers in [1,2k+1] which are quadratic residues mod p.

R. K. Guy asks whether there is an elementary proof for the fact that there are always less quadratic residues in the interval [2k+2,4k+2] than in [1,2k+1].

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = A104635(n) - A186709(n) = A186709(n) + A178154(n) = (A104635(n) + A178154(n))/2 = (A002145(n) + 2*A178154(n) - 1)/4.

Prog

(PARI) forprime( p=1, 499, p%4==3|next; u=3; c=[1, 0]; for(i=2, p-2, bittest(u, i^2%p) & next; u+=1<<(i^2%p); c[i^2%p*2\p+1]++); print1(c[1]", "))

Crossrefs

Cf. A002145, A104635, A186709, A178154.

Sequence in context: A282896 A141437 A219645 * A227697 A097457 A321612

Adjacent sequences: A186705 A186706 A186707 * A186709 A186710 A186711

Keyword

nonn

Author

M. F. Hasler, Feb 25 2011

Status

approved