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

1,1

Comments

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, ...

Links

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

Example

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.

Prog

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

Crossrefs

Cf. A303704, A330404.

Sequence in context: A205299 A222242 A069771 * A235801 A076986 A333386

Adjacent sequences: A330420 A330421 A330422 * A330424 A330425 A330426

Keyword

nonn

Author

Jianing Song, Dec 14 2019

Status

approved