A329095 Odd numbers k such that x^2 == 2 (mod k) has no solution.

3, 5, 9, 11, 13, 15, 19, 21, 25, 27, 29, 33, 35, 37, 39, 43, 45, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 85, 87, 91, 93, 95, 99, 101, 105, 107, 109, 111, 115, 117, 121, 123, 125, 129, 131, 133, 135, 139, 141, 143, 145, 147, 149, 153, 155, 157, 159, 163

Offset: 1

Author

Jianing Song, Nov 04 2019

Keywords

• nonn
• easy

Comments

Complement of A058529 over the odd numbers: odd numbers k such that x^2 == 2 (mod k) has solutions.

Odd numbers k such that at least one prime factor of k is congruent to 3 or 5 modulo 8 (at least one prime factor is in A003629).

Also odd terms in A025020.

Examples

			x^2 == 2 (mod 45) has no solution, so 45 is a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003629. A047621 is a subsequence.

Cf. A058529, A057126, A025020 (numbers k such that x^2 == 2 (mod k) has no solution).

Programs

Maple

filter:= proc(t) (numtheory:-factorset(t) mod 8) intersect {3,5} <> {} end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Nov 05 2019

Mathematica

Reap[Do[If[AnyTrue[FactorInteger[k][[All, 1]], MatchQ[Mod[#, 8], 3|5]&], Sow[k]], {k, 1, 999, 2}]][[2, 1]] (* Jean-François Alcover, Aug 22 2020 *)

PARI

isA329095(k) = (k%2) && !issquare(Mod(2,k))