A330404 Least nonsquare k that is a quadratic residue modulo n.

2, 2, 3, 5, 5, 3, 2, 8, 7, 5, 3, 12, 3, 2, 6, 17, 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, 2, 3, 21

Offset

1,1

Comments

a(n) >= n if and only if n is in A254328.

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, ...: 16, 48, 240, 288, 720, 720, 720, 720, 1008, 1440, ...

Links

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

Example

k is a quadratic residue modulo 16 if and only if k == 0, 1, 4, 9 (mod 16). Since 0, 1, 4, 9 and 16 are squares, a(16) = 17.
k is a quadratic residue modulo 48 if and only if k == 0, 1, 4, 9, 16, 25, 33, 36 (mod 48). Since 0, 1, 4, 9, 16 and 25 are squares, a(48) = 33.
k is a quadratic residue modulo 720 if and only if k == 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 145, ..., 676 (mod 720). Since 0, 1, 4, ..., 144 are squares, a(720) = 145.

Prog

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

Crossrefs

A309680 is an alternate version.

Cf. A254328, A330423.

Sequence in context: A360461 A284114 A323480 * A340575 A139171 A329570

Adjacent sequences: A330401 A330402 A330403 * A330405 A330406 A330407

Keyword

nonn

Author

Jianing Song, Dec 14 2019

Status

approved