A309680 The smallest nonsquare nonzero integer that is a quadratic residue modulo n, or 0 if no such integer exists.

0, 0, 0, 0, 0, 3, 2, 0, 7, 5, 3, 0, 3, 2, 6, 0, 2, 7, 5, 5, 7, 3, 2, 12, 6, 3, 7, 8, 5, 6, 2, 17, 3, 2, 11, 13, 3, 5, 3, 20, 2, 7, 6, 5, 10, 2, 2, 33, 2, 6, 13, 12, 6, 7, 5, 8, 6, 5, 3, 21, 3, 2, 7, 17, 10, 3, 6, 8, 3, 11, 2, 28, 2, 3, 6, 5, 11, 3, 2, 20, 7

Offset

1,6

Links

Table of n, a(n) for n=1..81.

Formula

a(n) = 2 for n in A057126 and n > 2. - Michel Marcus, Aug 24 2019

Example

For n=5, the nonzero quadratic residues modulo 5 are 1 and 4. Since these are both squares we have a(5) = 0.
For n=6, the nonzero quadratic residues modulo 6 are 1,3, and 4. Since 3 is not a square we have a(6) = 3.
For n=10, the nonzero quadratic residues modulo 10 are 1,4,5,6,9. Since 5 is the least nonsquare value, we have a(10) = 5.

Mathematica

a[n_] := SelectFirst[ Union@ Mod[Range[n-1]^2, n], ! IntegerQ@ Sqrt@ # &, 0]; Array[a, 81] (* Giovanni Resta, Aug 13 2019 *)

Prog

(PARI) a(n) = my(v=select(x->issquare(x), vector(n-1, k, Mod(k, n)), 1), y = select(x->!issquare(x), Vec(v))); if (#y, y[1], 0); \\ Michel Marcus, Aug 16 2019

Crossrefs

A330404 is an alternate version.

Cf. A046071, A053760, A057126.

Sequence in context: A080779 A355090 A319830 * A010604 A067585 A173787

Adjacent sequences: A309677 A309678 A309679 * A309681 A309682 A309683

Keyword

nonn

Author

John Prosser, Aug 12 2019

Status

approved