A005382 Primes p such that 2p-1 is also prime. (Formerly M0849)

2, 3, 7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011

Offset

1,1

Comments

Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - Benoit Cloitre, Jan 25 2002

Let q = 2n-1. For these n (and q), the sum of two cyclotomic polynomials can be written as a product of cyclotomic polynomials and as a cyclotomic polynomial in x^2: Phi(q,x) + Phi(2q,x) = 2 Phi(n,x) Phi(2n,x) = 2 Phi(n,x^2). - T. D. Noe, Nov 04 2003

Primes in A006254. - Zak Seidov, Mar 26 2013

If a(n) is in A168421 then A005383(n) is a twin prime with a Ramanujan prime, A005383(n) - 2. If this sequence has an infinite number of terms in A168421, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013

References

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..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.

R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Ajeet Kumar, Subhamoy Maitra, and Chandra Sekhar Mukherjee, On approximate real mutually unbiased bases in square dimension, Cryptography and Communications (2020) Vol. 13, 321-329.

Marius Tărnăuceanu, Arithmetic progressions in finite groups, arXiv:2003.10060 [math.GR], 2020.

Wikipedia, Cunningham chain

Formula

a(n) = A129521(n) / A005383(n). - Reinhard Zumkeller, Apr 19 2007

a(n) = (A005383(n) + 1)/2. - Zak Seidov, Nov 04 2010

Maple

f := proc(Q) local t1, i, j; t1 := []; for i from 1 to 500 do j := ithprime(i); if isprime(2*j-Q) then t1 := [op(t1), j]; fi; od: t1; end; f(1);

Mathematica

Select[Prime[Range[300]], PrimeQ[2#-1]&]

Prog

(Magma) [n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)] // Vincenzo Librandi, Nov 18 2010
(PARI) select(p->isprime(2*p-1), primes(500)) \\ Charles R Greathouse IV, Apr 26 2012
(Haskell)
a005382 n = a005382_list !! (n-1)
a005382_list = filter
   ((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
-- Reinhard Zumkeller, Oct 03 2012
(PARI) forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

Crossrefs

Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.

Cf. A010051, A000040, A053685 (subsequence), A006254.

Sequence in context: A334050 A073640 A174568 * A195354 A244638 A113165

Adjacent sequences: A005379 A005380 A005381 * A005383 A005384 A005385

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved