A330404 -id:A330404 - cp's OEIS frontend

A330423 Least nonsquare k that is a quadratic residue modulo n and is coprime to n.

2, 3, 7, 5, 6, 7, 2, 17, 7, 11, 3, 13, 3, 11, 19, 17, 2, 7, 5, 21, 22, 3, 2, 73, 6, 3, 7, 29, 5, 19, 2, 17, 31, 13, 11, 13, 3, 5, 10, 41, 2, 37, 6, 5, 19, 3, 2, 73, 2, 11, 13, 17, 6, 7, 14, 57, 7, 5, 3, 61, 3, 5, 22, 17, 14, 31, 6, 13, 13, 11, 2, 73, 2, 3, 19, 5, 15

Offset: 1

Jianing Song, Dec 14 2019

nonn

a(n) > n if and only if n is in A303704.

It seems that lim_{n->oo} a(n)/n = 0. Conjectured last term m such that a(m)/m > 1/k, k = 1, 2, 3, ...: 840, 1680, 2640, 9240, 10920, 10920, 18480, 18480, 21840, 29640, ...

			k is a coprime quadratic residue modulo 16 if and only if k == 1 (mod 8). Since 1 and 9 are squares, a(16) = 17.
k is a coprime quadratic residue modulo 24 if and only if k == 1 (mod 24). Since 1, 25 and 49 are squares, a(24) = 73.
k is a coprime quadratic residue modulo 840 if and only if k == 1, 121, 169, 289, 361, 529 (mod 840). Since 1, 121, 169, 289, 361, 529, 841, 961 are squares, a(840) = 840+169 = 1009.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A303704, A330404.

a(n) = my(k=1); while(!issquare(Mod(k,n)) || issquare(k) || gcd(k,n)>1, k++); k

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

Original entry on oeis.org

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

John Prosser, Aug 12 2019

nonn

			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.

A330404 is an alternate version.

Cf. A046071, A053760, A057126.

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

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

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

cp's OEIS Frontend

A330423 Least nonsquare k that is a quadratic residue modulo n and is coprime to n.

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