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

Author

Jonathan Sondow, May 16 2010

Keywords

• easy
• nonn

Comments

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

Examples

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

Links

• Wikipedia, Quadratic residue
• Wikipedia, Quadratic reciprocity

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)).

Programs

Mathematica

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

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

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.