%I A005384 M0731
%S A005384 2,3,5,11,23,29,41,53,83,89,113,131,173,179,191,233,239,251,281,293,359,
%T A005384 419,431,443,491,509,593,641,653,659,683,719,743,761,809,911,953,1013,
%U A005384 1019,1031,1049,1103,1223,1229,1289,1409,1439,1451,1481,1499,1511,1559
%N A005384 Sophie Germain primes p: 2p+1 is also prime.
%C A005384 Then 2p+1 is called a safe prime: see A005385.
%C A005384 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
%C A005384 Subsequence of A117360. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 10 2006
%C A005384 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
%C A005384 A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= 
               n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 
               18 2009]
%C A005384 a(n) mod 10 <> 7. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 12 2009]
%C A005384 Near subset of A161896. [From Reikku Kulon (reikku(AT)gmail.com), Jun 
               21 2009]
%C A005384 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]
%C A005384 Contribution from Daniel Forgues (squid(AT)zensearch.com), Jul 31 2009: 
               (Start)
%C A005384 A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e. p 
               = 5 (mod 6).
%C A005384 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)
%C A005384 Vincenzo Librandi's conjecture (above) follows from Pocklington's theorem. 
               [From T. D. Noe (noe(AT)sspectra.com), Aug 02 2009]
%D A005384 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.
%D A005384 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
%D A005384 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005384 T. D. Noe, <a href="b005384.txt">Table of n, a(n) for n = 1..10000</a>
%H A005384 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005384 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=SophieGermainPrime">Sophie Germain Prime</a>
%H A005384 Reikku Kulon, <a href="a005384.c">Sublinear arbitrary precision generator 
               of Sophie Germain and safe primes in C</a> (public domain)
%H A005384 H. Lifchitz, <a href="http://ourworld.compuserve.com/homepages/hlifchitz/
               Henri/us/ThSgus.htm">A new and simpler primality test for Sophie-Germain 
               numbers(q=2*p+1)</a>
%H A005384 Martin M. Musatov, <a href="http://www.knowledgerush.com/kr/encyclopedia/
               Sophie_Germain_prime/">Sophie Germain primes</a>
%H A005384 L. Riddle, <a href="http://www.agnesscott.edu/Lriddle/WOMEN/germain-FLT/
               SGandFLT.htm">Sophie Germain and Fermat's Last Theorem</a>
%H A005384 T. Tao, <a href="http://arXiv.org/abs/math.NT/0505402">Obstructions to 
               uniformity and arithmetic patterns in the primes</a>
%H A005384 Vmoraru, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               GermainPrime.html">Germain prime</a>
%H A005384 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SophieGermainPrime.html">Link to a section of The World of Mathematics.</
               a>
%H A005384 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A005384 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sophie_Germain_prime">
               Sophie Germain prime</a>
%p A005384 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
%t A005384 Select[Prime[Range[1000]], PrimeQ[2#+1]&]
%o A005384 (MAGMA) [ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; [From Klaus Brockhaus 
               (klaus-brockhaus(AT)t-online.de), Jan 01 2009]
%Y A005384 Cf. A005385, A007700, A023272, A023302, A023330, A057331, A005602.
%Y A005384 Cf. A087634.
%Y A005384 Cf. A000355, A156541, A156542, A156592. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 12 2009]
%Y A005384 Cf. A161896 [From Reikku Kulon (reikku(AT)gmail.com), Jun 21 2009]
%Y A005384 Sequence in context: A118332 A118333 A131101 this_sequence A118571 A118504 
               A038905
%Y A005384 Adjacent sequences: A005381 A005382 A005383 this_sequence A005385 A005386 
               A005387
%K A005384 nonn,nice
%O A005384 1,1
%A A005384 N. J. A. Sloane (njas(AT)research.att.com).