%I M4344 N1819 #265 Oct 09 2024 04:33:07

%S 7,13,19,31,37,43,61,67,73,79,97,103,109,127,139,151,157,163,181,193,

%T 199,211,223,229,241,271,277,283,307,313,331,337,349,367,373,379,397,

%U 409,421,433,439,457,463,487,499,523,541,547,571,577,601,607,613,619

%N Primes of the form 6m + 1.

%C Equivalently, primes of the form 3m + 1.

%C Rational primes that decompose in the field Q(sqrt(-3)). - _N. J. A. Sloane_, Dec 25 2017

%C Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - _Benoit Cloitre_, Feb 08 2003

%C Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - _Benoit Cloitre_, May 04 2003

%C Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - _N. J. A. Sloane_, May 31 2014

%C Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - _T. D. Noe_, May 07 2005

%C 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

%C A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - _Jonathan Vos Post_, Sep 03 2006

%C A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - _Reinhard Zumkeller_, Oct 21 2007

%C 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

%C 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

%C 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

%C 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

%C For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - _R. J. Mathar_, Apr 16 2024

%D 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.

%D David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ray Chandler, <a href="/A002476/b002476.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node18.html">Calcul de (-3/p)</a>

%H Barry Brent, <a href="http://math.colgate.edu/~integers/y18/y18.pdf">Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups</a>, Integers (2024) Vol. 24, Art. No. A18. See p. 13.

%H F. S. Carey, <a href="https://archive.org/stream/proceedingslond00unkngoog#page/n315/mode/2up">On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p</a>, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.

%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.

%H Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, <a href="https://doi.org/10.7546/nntdm.2024.30.3.516-529">The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences</a>, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

%H K. G. Reuschle, <a href="https://gdz.sub.uni-goettingen.de/id/PPN576716448?tify={%22pages%22:[9]}">Tafeln complexer Primzahlen</a>, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

%H Neville Robbins, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-4.pdf">On the Infinitude of Primes of the Form 3k+1</a>, Fib. Q., 43,1 (2005), 29-30.

%H 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 <a href="https://oeis.org/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%F From _R. J. Mathar_, Apr 03 2011: (Start)

%F Sum_{n >= 1} 1/a(n)^2 = A175644.

%F Sum_{n >= 1} 1/a(n)^3 = A175645. (End)

%F a(n) = 6*A024899(n) + 1. - _Zak Seidov_, Aug 31 2016

%F From _Vaclav Kotesovec_, May 02 2020: (Start)

%F Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.

%F Product_{k>=1} (1 + 1/a(k)^2) = A334481.

%F Product_{k>=1} (1 - 1/a(k)^3) = A334478.

%F Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)

%F Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - _Wolfdieter Lang_, Mar 03 2021

%e 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)).)

%e Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.

%e 17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

%p a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

%t Select[6*Range[100] + 1, PrimeQ[ # ] &] (* _Stefan Steinerberger_, Apr 06 2006 *)

%o (Magma) [n: n in [1..700 by 6] | IsPrime(n)]; // _Vincenzo Librandi_, Apr 05 2011

%o (PARI) select(p->p%3==1,primes(100)) \\ _Charles R Greathouse IV_, Oct 31 2012

%o (Haskell)

%o a002476 n = a002476_list !! (n-1)

%o a002476_list = filter ((== 1) . (`mod` 6)) a000040_list

%o -- _Reinhard Zumkeller_, Jan 15 2013

%o (J) (#~ 1&p:) >: 6 * i.1000 NB. _Stephen Makdisi_, May 01 2018

%o (GAP) Filtered(List([0..110],k->6*k+1),n-> IsPrime(n)); # _Muniru A Asiru_, Mar 11 2019

%Y Cf. A045331, A242660.

%Y For values of m see A024899. Primes of form 3n - 1 give A003627.

%Y These are the primes arising in A024892, A024899, A034936.

%Y A091178 gives prime index.

%Y Cf. A006512, A007528.

%Y Subsequence of A016921 and of A050931.

%Y Cf. A004611 (multiplicative closure).

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_