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 A003627 Primes of the form 3n-1. (Formerly M1388) 82
 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017 Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011 Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003 A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007 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 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 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 Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011 A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011 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 Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012 For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017 2 adjoined to A007528. - David A. Corneth, Nov 12 2017 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 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. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 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]. A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004. Eric Weisstein's World of Mathematics, Eisenstein Prime Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317. FORMULA From R. J. Mathar, Apr 03 2011: (Start) Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644. Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End) MAPLE 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); MATHEMATICA Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *) PROG (MAGMA) [n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011 (Haskell) a003627 n = a003627_list !! (n-1) a003627_list = filter ((== 2) . (`mod` 3)) a000040_list -- Reinhard Zumkeller, Oct 30 2011 (PARI) is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013 CROSSREFS Primes of form 3n+1 give A002476. These are the primes arising in A024893, A087370, A088879. A091177 gives prime index. Cf. A001359, A007528, A007645, A221717, A057145. Subsequence of A034020. Sequence in context: A164921 A156830 A140556 * A103203 A105875 A031368 Adjacent sequences:  A003624 A003625 A003626 * A003628 A003629 A003630 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)