A003627 Primes of the form 3n-1.

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

Offset: 1

Author

N. J. A. Sloane and Mira Bernstein

Keywords

• nonn
• easy

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.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• 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.
• Index to sequences related to decomposition of primes in quadratic fields

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.

Programs

Haskell

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

Magma

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 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

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

PARI

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

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)