1,1

Then 2p+1 is called a safe prime: see A005385.

Primes such that the equation phi(k) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe (noe(AT)sspectra.com), Oct 24 2003

Subsequence of A117360. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 10 2006

Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe (noe(AT)sspectra.com), Jan 04 2008

A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 18 2009]

a(n) mod 10 <> 7. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 12 2009]

Near subset of A161896. [From Reikku Kulon (reikku(AT)gmail.com), Jun 21 2009]

Conjecture: let p = prime numbers 2*p+1 is prime if and only if 2^(2p)-1 = 0 mod. (2p+1) [From Vincenzo Libandi (vincenzo.librandi(AT)tin.it), Jul 23 2009]

Contribution from Daniel Forgues (squid(AT)zensearch.com), Jul 31 2009: (Start)

A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e. p = 5 (mod 6).

A prime p of the form 6k+1, k >= 1, i.e. p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. (End)

Vincenzo Librandi's conjecture (above) follows from Pocklington's theorem. [From T. D. Noe (noe(AT)sspectra.com), Aug 02 2009]

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.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 [alternative scanned copy].

C. K. Caldwell, The Prime Glossary, Sophie Germain Prime

Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain)

H. Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1)

Martin M. Musatov, Sophie Germain primes

L. Riddle, Sophie Germain and Fermat's Last Theorem

T. Tao, Obstructions to uniformity and arithmetic patterns in the primes

Vmoraru, PlanetMath.org, Germain prime

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Wikipedia, Sophie Germain prime

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1)=true then A:=A union {ithprime(n)} else A:=A fi od: A:=A; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 09 2004

Select[Prime[Range[1000]], PrimeQ[2#+1]&]

(MAGMA) [ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 01 2009]

Cf. A005385, A007700, A023272, A023302, A023330, A057331, A005602.

Cf. A087634.

Cf. A000355, A156541, A156542, A156592. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 12 2009]

Cf. A161896 [From Reikku Kulon (reikku(AT)gmail.com), Jun 21 2009]

Sequence in context: A118332 A118333 A131101 this_sequence A118571 A118504 A038905

Adjacent sequences: A005381 A005382 A005383 this_sequence A005385 A005386 A005387

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).