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URL: https://oeis.org/A141171

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A141171
Primes of the form -x^2+4*x*y+2*y^2 (as well as of the form 5*x^2+8*x*y+2*y^2).
7
2, 5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 431, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 911, 941, 983, 1013, 1031, 1061, 1103, 1109, 1151, 1181, 1223, 1229, 1277, 1301, 1319, 1367
OFFSET
1,1
COMMENTS
Discriminant is 24. Class is 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Also primes of form 6*u^2 - v^2. The transformation {u, v} = {y, x - 2*y} yields the form in the title. - Juan Arias-de-Reyna, Mar 19 2011
Members of A141171 but not of A105880: 2, 431, 911, 1013, 1181, ..., . - Robert G. Wilson v, Aug 30 2013
This is also the list of primes p such that p = 2 or p is congruent to 5 or 23 mod 24 - Jean-François Alcover, Oct 28 2016
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
D. B. Zagier, Zetafunktionen und quadratische Körper.
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
EXAMPLE
a(4) = 29 because we can write 29 = -1^2 + 4*1*3 + 2*3^2 (or 29 = 5*1^2 + 8*1*2 + 2*2^2).
MAPLE
N:= 10^5: # to get all terms <= N
select(t -> isprime(t) and [isolve(6*u^2-v^2=t)]<>[], [2, seq(op([24*i+5, 24*i+23]), i=0..floor((N-5)/24))]); # Robert Israel, Sep 28 2014
MATHEMATICA
A141171 = {}; Do[p = -x^2 + 4 * x * y + 2 * y^2; If[p > 0 && PrimeQ@ p, AppendTo[A141171, p]], {x, 25}, {y, 25}]; Take[ Union@ A141171, 57] (* Robert G. Wilson v, Aug 30 2013 *)
Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 24], 5|23]&] (* Jean-François Alcover, Oct 28 2016 *)
CROSSREFS
Cf. A141170 (d = 24), A105880 (Primes for which -8 is a primitive root.) A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Cf. also A242665.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Sequence in context: A355025 A198444 A019368 * A180535 A038919 A141181
KEYWORD
nonn
AUTHOR
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008
STATUS
approved