cp's OEIS Frontend

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).

%I A007645 M2637 #113 Feb 16 2025 08:32:31
%S A007645 3,7,13,19,31,37,43,61,67,73,79,97,103,109,127,139,151,157,163,181,
%T A007645 193,199,211,223,229,241,271,277,283,307,313,331,337,349,367,373,379,
%U A007645 397,409,421,433,439,457,463,487,499,523,541,547,571,577,601,607,613
%N 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).
%C A007645 Also, odd primes p such that -3 is a square mod p. - _N. J. A. Sloane_, Dec 25 2017
%C A007645 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.
%C A007645 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
%C A007645 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
%C A007645 Conjecture: this sequence is Union(A002383,A162471). - _Daniel Tisdale_, Jul 04 2009
%C A007645 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
%C A007645 Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - _Reinhard Zumkeller_, Oct 30 2011
%C A007645 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
%C A007645 The prime factors of A002061. - _Richard R. Forberg_, Dec 10 2014
%C A007645 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 A007645 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
%D A007645 Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
%D A007645 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
%D A007645 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007645 Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.
%H A007645 T. D. Noe, <a href="/A007645/b007645.txt">Table of n, a(n) for n = 1..1000</a>
%H A007645 U. P. Nair, <a href="http://arXiv.org/abs/math.NT/0408107">Elementary results on the binary quadratic form a^2+ab+b^2</a>, arXiv:math/0408107 [math.NT], 2004.
%H A007645 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H A007645 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EisensteinInteger.html">Eisenstein Integer.</a>
%F A007645 p == 0 or 1 (mod 3).
%F A007645 {3} UNION A002476. - _R. J. Mathar_, Oct 28 2008
%F A007645 A007645 UNION A003627 = A000040. - _Juri-Stepan Gerasimov_, Jan 28 2010
%p A007645 select(isprime,[3, seq(6*k+1, k=1..1000)]); # _Robert Israel_, Dec 12 2014
%t A007645 Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* _Harvey P. Dale_, Aug 21 2021 *)
%o A007645 (PARI) forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ _Charles R Greathouse IV_, Jun 16 2011
%o A007645 (Haskell)
%o A007645 a007645 n = a007645_list !! (n-1)
%o A007645 a007645_list = filter ((== 1) . a010051) $ tail a003136_list
%o A007645 -- _Reinhard Zumkeller_, Jul 11 2013, Oct 30 2011
%Y A007645 Subsequence of A003136.
%Y A007645 Subsequences include A002407, A002648, and A201477.
%Y A007645 Apart from initial term, same as A045331.
%Y A007645 Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
%Y A007645 Cf. A000040, A003627.
%Y A007645 Primes in A003136 and A034017.
%K A007645 nonn,easy
%O A007645 1,1
%A A007645 _N. J. A. Sloane_, _Mira Bernstein_ and _Robert G. Wilson v_
%E A007645 Entry revised by _N. J. A. Sloane_, Jan 29 2013