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).

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

Offset: 1

Author

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

Keywords

• nonn
• easy

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(-3|p) = +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. Springer-Verlag, NY, 1989, p. 50.
• 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.

Crossrefs

Subsequence of A003136.

Subsequences include A002407, A002648, and A201477.

Apart from initial term, same as A045331.

Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).

Cf. A000040, A003627.

Primes in A003136 and A034017.

Programs

Haskell

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

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 *)

PARI

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

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

Extensions

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