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A007645
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Cuban primes: primes of the form x^2 + xy + y^2; or: primes of form x^2 + 3*y^2; or: primes == 0 or 1 mod 3.
(Formerly M2637)
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58
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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 (njas(AT)research.att.com), 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 (noe(AT)sspectra.com), May 19 2008
Conjecture: this sequence is Union(A002383,A162471). [From Daniel Tisdale (daniel6874(AT)gmail.com), Jul 04 2009]
Primes of the form 3*m+1/2-+1/2. Or primes of the form 6*k-1-+2. A007645 UNION A003627 = A000040. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 28 2010]
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. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 01 2010]
Subsequence of Loeschian numbers, cf. A003136; A088534(a(n)) > 0. [Reinhard Zumkeller, Oct 30 2011]
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REFERENCES
| Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
D. Cox, "Primes of 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.
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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
Eric Weisstein's World of Mathematics, Eisenstein Integer.
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FORMULA
| p == 0 or 1 mod 3.
{3} UNION A002476. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 28 2008]
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MATHEMATICA
| Clear[f, lst, p, x, y]; f[x_, y_]:=x^2+x*y+y^2; lst={}; Do[Do[p=f[x, y]; If[PrimeQ[p]&&p<3614, AppendTo[lst, p]], {y, 0, 3*5!}], {x, 0, 3*5!}]; Take[Union[lst], 250] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 04 2009]
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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) a003136_list
-- Reinhard Zumkeller, Oct 30 2011
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CROSSREFS
| Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2)
Cf. A000040, A003627. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 28 2010]
Sequence in context: A086148 A205956 A167462 * A144919 A015916 A023203
Adjacent sequences: A007642 A007643 A007644 * A007646 A007647 A007648
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein and Robert G. Wilson v (rgwv(AT)rgwv.com)
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