A177864 a(n) is the smallest nontrivial quadratic residue modulo prime(n), for n >= 3.

4, 2, 3, 3, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 4, 4, 2, 3, 4, 2, 4, 3, 3, 2, 2, 4, 2, 4, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 4, 4, 4, 2, 2, 4, 4, 2, 3, 3, 2, 2, 2, 3, 4, 2, 4, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 3, 4, 2, 4, 2, 4, 3

Offset

3,1

Comments

There is no quadratic residue > 1 modulo the first or 2nd prime, so the sequence begins with a(3).

Links

Table of n, a(n) for n=3..100.

Wikipedia, Quadratic residue

Wikipedia, Quadratic reciprocity

Formula

a(n) = 2 or 3 or 4 according as prime(n) == 1,7,9,15,17,23 or 11,13 or 3,5,19,21 (mod 24), respectively, for n > 2, by the quadratic reciprocity law and its supplements.

Example

The quadratic residues modulo prime(3) = 5 are 1 and 4, so a(3) = 4.

Mathematica

Flatten[Table[ Extract[Flatten[ Position[Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]], {2}], {n, 3, 100}]]

Prog

(PARI) a(n, p=prime(n))=[2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2][p%24] \\ Charles R Greathouse IV, Jun 14 2022

Crossrefs

Cf. A063987 (triangle in which the n-th row gives the quadratic residues modulo prime(n)), A053760 (smallest positive quadratic nonresidue modulo prime(n)).

Sequence in context: A307367 A136626 A079623 * A090112 A226087 A118945

Adjacent sequences: A177861 A177862 A177863 * A177865 A177866 A177867

Keyword

easy,nonn

Author

Jonathan Sondow, May 16 2010

Status

approved