1,1

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

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.

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.

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

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

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

(Haskell)

a007645 n = a007645_list !! (n-1)

a007645_list = filter ((== 1) . a010051) $ tail a003136_list

-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

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 and A034017.

Sequence in context: A262086 A205956 A215907 * A144919 A215801 A215809

Adjacent sequences: A007642 A007643 A007644 * A007646 A007647 A007648

nonn,easy

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

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

approved