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Search: id:A007645
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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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+0
41
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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; listen)
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OFFSET
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1,1
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COMMENT
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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]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, p. 7.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
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
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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
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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
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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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CROSSREFS
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Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2)
Sequence in context: A099957 A086148 A167462 this_sequence A144919 A015916 A023203
Adjacent sequences: A007642 A007643 A007644 this_sequence A007646 A007647 A007648
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein and Robert G. Wilson v (rgwv(AT)rgwv.com)
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