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A141173 Primes of the form -2*x^2+2*x*y+3*y^2 (as well as of the form 6*x^2+10*x*y+3*y^2). 7
3, 7, 19, 31, 47, 59, 83, 103, 131, 139, 167, 199, 223, 227, 251, 271, 283, 307, 311, 367, 383, 419, 439, 467, 479, 503, 523, 563, 587, 607, 619, 643, 647, 691, 719, 727, 787, 811, 839, 859, 887, 971, 983, 1039, 1063, 1091, 1123, 1151, 1223, 1231, 1259, 1279, 1291, 1307, 1319, 1399, 1427 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Discriminant = 28. Class = 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 7*u^2-v^2. The transformation {u,v}={-x-y,3*x+2*y} yields the form in the title. [Juan Arias-de-Reyna, Mar 19 2011]

This is also the list of primes p such that p = 7 or p is congruent to 3, 19 or 27 mod 28 - Jean-François Alcover, Oct 28 2016

REFERENCES

Borevich and Shafaewich, Number Theory.

D. B. Zagier, Zetafunktionen und quadratische Koerper.

LINKS

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

EXAMPLE

a(3)=19 because we can write 19=-2*4^2+2*4*3+3*3^2 (or 19=6*1^2+10*1*1+3*1^2)

MATHEMATICA

Select[Prime[Range[250]], # == 7 || MatchQ[Mod[#, 28], 3|19|27]&] (* Jean-François Alcover, Oct 28 2016 *)

CROSSREFS

Cf. A141172 (d=28) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).

Cf. also A242666.

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: A240072 A066148 A093932 * A145472 A217199 A077313

Adjacent sequences:  A141170 A141171 A141172 * A141174 A141175 A141176

KEYWORD

nonn

AUTHOR

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

STATUS

approved

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Last modified April 25 12:04 EDT 2019. Contains 322456 sequences. (Running on oeis4.)