

A007645


Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).
(Formerly M2637)


86



3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613
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OFFSET

1,1


COMMENTS

Also, odd primes p such that 3 is a square mod p.  N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3  y^3)/(x  y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (1+sqrt(3))/2. The present sequence gives the rational primes which are also Eisenstein primes.  N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633.  T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471).  Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p  1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934.  Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0.  Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p).  Jon Perry, Feb 02 2014
The prime factors of A002061.  Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == 3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = 3). p = 2 is not a solution. x^2 == 3 (mod 4) has solutions for all odd x. x^2 == 3 (mod p) has for odd primes p, not 3, the solutions of Legendre(3p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == 3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)).  Wolfdieter Lang, May 22 2021


REFERENCES

D. A. Buell, Binary Quadratic Forms. SpringerVerlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 220223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319323, 1991.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.


FORMULA

p == 0 or 1 (mod 3).
{3} UNION A002476.  R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040.  JuriStepan Gerasimov, Jan 28 2010


MAPLE

select(isprime, [3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014


MATHEMATICA

Join[{3}, Select[Prime[Range[150]], Mod[#, 3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)


PROG

(PARI) forprime(p=2, 1e3, if(p%3<2, print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a007645 n = a007645_list !! (n1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
 Reinhard Zumkeller, Jul 11 2013, Oct 30 2011


CROSSREFS

Apart from initial term, same as A045331.
Cf. A002407 (cuban primes, a subsequence). A002648 and A201477 are also subsequences.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2)
Cf. A000040, A003627.  JuriStepan Gerasimov, Jan 28 2010
Primes in A003136 and A034017.
Sequence in context: A262086 A205956 A215907 * A144919 A215801 A215809
Adjacent sequences: A007642 A007643 A007644 * A007646 A007647 A007648


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v


EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 29 2013


STATUS

approved



