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 A003627 Primes of the form 3n-1. (Formerly M1388) 82

%I M1388

%S 2,5,11,17,23,29,41,47,53,59,71,83,89,101,107,113,131,137,149,167,173,

%T 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,

%U 383,389,401,419,431,443,449,461,467,479,491,503,509,521,557,563,569,587

%N Primes of the form 3n-1.

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

%C Primes p such that 1+x+x^2 is irreducible over GF(p). - _Joerg Arndt_, Aug 10 2011

%C Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - _Benoit Cloitre_, Feb 08 2003

%C A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - _Reinhard Zumkeller_, Oct 21 2007

%C The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - _Paul Muljadi_, Jun 05 2008

%C Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - _Daniel Tisdale_, Jul 04 2009

%C Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - _Juri-Stepan Gerasimov_, Jan 28 2010

%C Except for first term "2", all these prime numbers are of the form: 6*n-1. - _Vladimir Joseph Stephan Orlovsky_, Jul 13 2011

%C A088534(a(n)) = 0. - _Reinhard Zumkeller_, Oct 30 2011

%C For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - _Gary Detlefs_, Jan 02 2012

%C Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - _Gary Detlefs_, May 06 2012

%C For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - _Michel Marcus_, Nov 12 2017

%C 2 adjoined to A007528. - _David A. Corneth_, Nov 12 2017

%C For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - _Ralf Steiner_, May 17 2018

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

%H T. D. Noe, <a href="/A003627/b003627.txt">Table of n, a(n) for n = 1..1000</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinPrime.html">Eisenstein Prime</a>

%H Kenneth S. Williams, <a href="http://www.mscand.dk/article/view/11555/9571">3 as a Ninth Power (mod p)</a>, Math. Scand., Vol 35 (1974), 309-317.

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_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 = 0.30792... = A085548 - 1/9 - A175644.

%F Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

%p t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

%t Select[Range[-1, 600, 3], PrimeQ[#] &] (* _Vincenzo Librandi_, Jun 17 2015 *)

%o (MAGMA) [n: n in PrimesUpTo(720) | n mod 3 eq 2]; // _Bruno Berselli_, Apr 05 2011

%o a003627 n = a003627_list !! (n-1)

%o a003627_list = filter ((== 2) . (`mod` 3)) a000040_list

%o -- _Reinhard Zumkeller_, Oct 30 2011

%o (PARI) is(n)=n%3==2 && isprime(n) \\ _Charles R Greathouse IV_, Mar 20 2013

%Y Primes of form 3n+1 give A002476.

%Y These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.

%Y Cf. A001359, A007528, A007645, A221717, A057145.

%Y Subsequence of A034020.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_ and _Mira Bernstein_

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)