|
%I M0731
%S 2,3,5,11,23,29,41,53,83,89,113,131,173,179,191,233,239,251,281,293,
%T 359,419,431,443,491,509,593,641,653,659,683,719,743,761,809,911,953,
%U 1013,1019,1031,1049,1103,1223,1229,1289,1409,1439,1451,1481,1499,1511,1559
%N Sophie Germain primes p: 2p+1 is also prime.
%C Then 2p+1 is called a safe prime: see A005385.
%C Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - _T. D. Noe_, Oct 24 2003
%C Subsequence of A117360. - _Reinhard Zumkeller_, Mar 10 2006
%C 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_, Jan 04 2008
%C A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. [_Reinhard Zumkeller_, Feb 18 2009]
%C a(n) mod 10 <> 7. [_Reinhard Zumkeller_, Feb 12 2009]
%C Near subset of A161896. [_Reikku Kulon_, Jun 21 2009]
%C Contribution from _Daniel Forgues_, Jul 31 2009: (Start)
%C A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e. p = 5 (mod 6).
%C 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 Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - _Enrique Pérez Herrero_, May 03 2012
%C Solutions of the equation n'+(2n+1)'=2, where n' is the arithmetic derivative of n. - _Paolo P. Lava_, Aug 10 2012.
%C tau(4*a(n) + 2) = tau(4*a(n)) - 2 for n > 1. - _Arkadiusz Wesolowski_, Aug 25 2012
%C eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2 for n > 1. - _Arkadiusz Wesolowski_, Aug 26 2012
%C p and 2p+1 are Sophie Germain primes if and only if p is prime and 2^(2p) == 1 (mod 2p+1). - _Vincenzo Librandi_, Oct 09 2012
%C In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number sum_{k=m}^n a(k)*b^(n-k) is prime. [_Zhi-Wei Sun_, Mar 26, 2013]
%D 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 H. Dubner, Large Sophie Germain Primes, Math. Comp., 65 (1996), 393-396
%D F. Oort, Prime numbers, 2013, http://www.math.uu.nl/~oort0109/Taiwan-PrimeNu2012-2013.pdf
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A. Takashi, On Sophie Germain primes. Number theory (Liptovský Ján, 1999). Tatra Mt. Math. Publ. 20 (2000), 65-73.
%D A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, 11:1-3 (1987) 81-92
%D S. Yates, Sophie Germain primes. In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991. pp. 882-886
%H J. S. Cheema, <a href="/A005384/b005384.txt">Table of n, a(n) for n = 1..100000</a>. [This replaces an earlier b-file computed by T. D. Noe]
%H 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 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=SophieGermainPrime">Sophie Germain Prime</a>
%H Reikku Kulon, <a href="/A005384/a005384.c">Sublinear arbitrary precision generator of Sophie Germain and safe primes in C</a> (public domain)
%H 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 L. Riddle, <a href="http://www.agnesscott.edu/Lriddle/WOMEN/germain-FLT/SGandFLT.htm">Sophie Germain and Fermat's Last Theorem</a>
%H T. Tao, <a href="http://arXiv.org/abs/math.NT/0505402">Obstructions to uniformity and arithmetic patterns in the primes</a>
%H Vmoraru, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/GermainPrime.html">Germain prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>
%H Larry Riddle, <a href="http://www.agnesscott.edu/lriddle/women/germain-flt/sgandflt.htm">Sophie Germain and Fermat's Last Theorem</a>, Agnes Scott College, Math. Dept.Jul, 1999.
%H Del Centina, Andrea. <a href="http://web.unife.it/progetti/geometria/Germain.html">Letters of Sophie Germain preserved in Florence</a> Historia Mathematica, Vol. 32 (2005), 60-75
%p 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, Dec 09 2004
%t Select[Prime[Range[1000]], PrimeQ[2#+1]&]
%o (MAGMA) [ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; [Klaus Brockhaus, Jan 01 2009]
%o (PARI) select(p->isprime(2*p+1),primes(1000)) \\ Newer versions
%o select(primes(1000),p->isprime(2*p+1)) \\ v. 2.4.2 and older
%Y Cf. A005385, A007700, A023272, A023302, A023330, A057331, A005602, A087634.
%Y Cf. A000355, A156541, A156542, A156592.
%Y Cf. A161896.
%Y Cf. A156660, A156874, A092816.
%Y Cf. A023212.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_.
|
