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 A002476 Primes of the form 6m + 1. (Formerly M4344 N1819) 192
 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, 619 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Equivalently, primes of the form 3m + 1. Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017 Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003 Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003 Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014 Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005 Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006 A006512 larger than 5 (Greater of twin primes) is a subset of this. - Jonathan Vos Post, Sep 03 2006 A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007 Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008 Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012 Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016 For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016 REFERENCES M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870. David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. C. Banderier, Calcul de (-3/p) F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312. A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004. K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1. Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30. N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) FORMULA From R. J. Mathar, Apr 03 2011: (Start) Sum_{n >= 1} 1/a(n)^2 = A175644. Sum_{n >= 1} 1/a(n)^3 = A175645. (End) a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016 From Vaclav Kotesovec, May 02 2020: (Start) Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646. Product_{k>=1} (1 + 1/a(k)^2) = A334481. Product_{k>=1} (1 - 1/a(k)^3) = A334478. Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End) EXAMPLE Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).) Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence. 17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence. MAPLE a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n]; MATHEMATICA Select[6*Range + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *) PROG (MAGMA) [n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011 (PARI) select(p->p%3==1, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012 (Haskell) a002476 n = a002476_list !! (n-1) a002476_list = filter ((== 1) . (`mod` 6)) a000040_list -- Reinhard Zumkeller, Jan 15 2013 (J) (#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018 (GAP) Filtered(List([0..110], k->6*k+1), n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019 CROSSREFS Cf. A045331, A242660. For values of m see A024899. Primes of form 3n - 1 give A003627. These are the primes arising in A024892, A024899, A034936. A091178 gives prime index. Cf. A006512. Subsequence of A016921 and of A050931. Cf. A004611 (multiplicative closure). Sequence in context: A218146 A129389 A107925 * A123365 A144921 A272409 Adjacent sequences:  A002473 A002474 A002475 * A002477 A002478 A002479 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified July 2 11:54 EDT 2020. Contains 335398 sequences. (Running on oeis4.)