A063908 Numbers k such that k and 2*k-3 are primes.

3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 127, 137, 157, 167, 181, 191, 193, 211, 223, 233, 241, 251, 263, 283, 311, 317, 331, 347, 373, 421, 431, 433, 443, 457, 461, 487, 521, 547, 563, 577, 587, 613, 617, 631, 641, 643, 647

Offset: 1

Author

N. J. A. Sloane, Aug 31 2001

Keywords

• nonn

Comments

If p is in this sequence then the products of positive powers of 3, p and 2p-3 are entries in A086486. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003

Median prime of AP3's starting at 3, i.e., triples of primes (3,p,q) in arithmetic progression. - M. F. Hasler, Sep 24 2009

a(n) = sum of the coprimes(p) mod (p+1). - J. M. Bergot, Nov 13 2014

A010051(2*a(n)-3) = 1. - Reinhard Zumkeller, Jul 02 2015

A098090 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

Examples

			From _K. D. Bajpai_, Nov 29 2019: (Start)
a(5) = 13 is prime and 2*13 - 3 = 23 is also prime.
a(6) = 17 is prime and 2*17 - 3 = 31 is also prime.
(End)
		

Links

• Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• Carlos Rivera, Puzzle 34.- Prime Triplets in arithmetic progression, The Prime Puzzles & Problems Connection. [From _M. F. Hasler_, Sep 24 2009]

Crossrefs

Cf. A005382.

Cf. A000040, A010051, A088878, A172287.

Cf. A259730.

Programs

Haskell

a063908 n = a063908_list !! (n-1)
a063908_list = filter
   ((== 1) . a010051' . (subtract 3) . (* 2)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

Magma

[n : n in [0..700] | IsPrime(n) and IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 14 2014

Maple

select(k -> andmap(isprime, [k, 2*k-3]), [seq(k, k=1.. 10^4)]); # K. D. Bajpai, Nov 29 2019

Mathematica

Select[Prime[Range[6! ]],PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

PARI

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); if (isprime(2*p - 3), write("b063908.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 02 2009

PARI

forprime( p=1,default(primelimit), isprime(2*p-3) && print1(p",")) \\ M. F. Hasler, Sep 24 2009

Formula

a(n) = A241817(n)/2. - Wesley Ivan Hurt, Apr 08 2018