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A003627 Primes of form 3n-1.
(Formerly M1388)
62
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

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). [From 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. [From 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. [From 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

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

Eric Weisstein's World of Mathematics, Eisenstein Prime

FORMULA

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. - R. J. Mathar, Apr 03 2011

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

lst={}; Do[If[PrimeQ[p=3*n-1], (*Print[p]; *)AppendTo[lst, p]], {n, 10^3}]; lst [From Vladimir Orlovsky, Aug 21 2008]

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, A007645, A221717.

Subsequence of A034020.

Sequence in context: A164921 A156830 A140556 * A103203 A105875 A031368

Adjacent sequences:  A003624 A003625 A003626 * A003628 A003629 A003630

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane and Mira Bernstein

STATUS

approved

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Last modified July 23 07:46 EDT 2014. Contains 244850 sequences.