A139513 Primes congruent to {1, 3, 7, 9} mod 20.

3, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563, 569, 587, 601, 607, 641

Offset: 1

Author

Artur Jasinski, Apr 25 2008

Keywords

• nonn
• easy

Comments

Rational primes that decompose in the field Q(sqrt(-5)). - N. J. A. Sloane, Dec 25 2017

References

• Dirichlet & Dedekind, Lectures on Number Theory (English Translation 1999), p. 119.
• David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989, p. 14 (1.8), p. 32 (2.19).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A296922, A003626.

Programs

Magma

[ p: p in PrimesUpTo(700) | p mod 20 in [1,3,7,9] ]; // Vincenzo Librandi, Aug 15 2012

Mathematica

a = {}; Do[If[MemberQ[{1, 3, 7, 9}, Mod[Prime[n], 20]], AppendTo[a, Prime[n]]], {n, 1, 200}]; a (*Artur Jasinski*)
Select[Prime[Range[200]],MemberQ[{1,3,7,9},Mod[#,20]]&] (* Vincenzo Librandi, Aug 15 2012 *)

PARI

select(p->my(k=p%20); k==1 || k==3 || k==7 || k==9, primes(100)) \\ Charles R Greathouse IV, Nov 29 2016

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

Legendre symbol (-5, a(n)) = +1. One sets (-5, 5) = 0 and for odd primes p == -1, -3, -7, -9 (mod 20) (-5, p) = -1, given in A003626. - Wolfdieter Lang, Mar 05 2021