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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)
81
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also, 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

REFERENCES

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 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. 319-323, 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. - Juri-Stepan 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[500]], MemberQ[{1}, Mod[#, 3]] &]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)

PROG

(PARI) forprime(p=2, 1e3, if(p%3<2, print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

(Haskell)

a007645 n = a007645_list !! (n-1)

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. - Juri-Stepan Gerasimov, Jan 28 2010

Primes in A003136.

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

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Last modified May 20 13:54 EDT 2018. Contains 304338 sequences. (Running on oeis4.)