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A141159 Duplicate of A139492. 9
7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Name was: Primes of the form x^2 + 3*x*y - 3*y^2 (as well as of the form x^2 + 5*x*y + y^2).

Discriminant = 21. Class number = 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 (primitive).

Primes of the form 6n+1 which cannot be expressed as 7k-1, 7k-2, or 7k-4. a(n)^2 == 1 (mod 24). - Gary Detlefs, Jan 26 2014

Besides 7 (which divides 21), primes of the form  p == 1 (mod 3) and either == 1 or 2 or 4 (mod 7). For the other class, the primes represented by the principal form [3, 3, -1] (or primitive forms equivalent to this) are besides 3 (which divides 21), congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). For the primes of both classes see A038893. - Wolfdieter Lang, Jun 19 2019

REFERENCES

Z. I. Borevich and I. R. Shafarevich, Number Theory.

D. B. Zagier, Zetafunktionen und quadratische Koerper.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Peter Luschny, Binary Quadratic Forms

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

EXAMPLE

a(1)=7 because we can write 7 = 2^2 + 3*2*1 - 3*1^2 (or 7 = 1^2 + 5*1*1 + 1^2).

MAPLE

f:=n->7*ceil((6*n+1)/7)-(6*n+1):for n from 1 to 220 do if isprime(6*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(6*n+1) fi od. # Gary Detlefs, Jan 26 2014

MATHEMATICA

xy[{x_, y_}]:={x^2+3x y-3y^2, y^2+3x y -3x^2}; Union[Select[Flatten[xy/@ Subsets[ Range[50], {2}]], #>0&&PrimeQ[#]&]] (* Harvey P. Dale, Feb 17 2013 *)

PROG

(Sage) # uses[binaryQF]

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.

Q = binaryQF([1, 3, -3])

Q.represented_positives(1326, 'prime') # Peter Luschny, Jun 24 2019

CROSSREFS

Sequence in context: A043010 A322174 A139492 * A092475 A106924 A076285

Adjacent sequences:  A141156 A141157 A141158 * A141160 A141161 A141162

KEYWORD

dead

AUTHOR

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jun 12 2008

EXTENSIONS

More terms from Harvey P. Dale, Feb 17 2013

STATUS

approved

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)