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A005384 Sophie Germain primes p: 2p+1 is also prime.
(Formerly M0731)
296
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

Subsequence of A117360. - Reinhard Zumkeller, 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, Jan 04 2008

A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009

Near subset of A161896. - Reikku Kulon, Jun 21 2009

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. - Daniel Forgues, Jul 31 2009

Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012

Solutions of the equation n'+(2n+1)'=2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Aug 10 2012.

tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012

eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012

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

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

If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013

First member of a prime pair (p, q) in a 2p+1 progression (q = prime = 2p + 1). Numbers p*q are in A156592. - Jaroslav Krizek, Nov 19 2013

REFERENCES

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.

A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, 11:1-3 (1987) 81-92

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).

A. Takashi, On Sophie Germain primes. Number theory (Liptovský Ján, 1999). Tatra Mt. Math. Publ. 20 (2000), 65-73.

S. Yates, Sophie Germain primes.  In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991.  pp. 882-886

LINKS

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]

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].

P. Bruillard, S.-H. Ng, E. Rowell, Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050, 2013

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

Del Centina, Andrea. Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), 60-75

H. Dubner, Large Sophie Germain Primes, Math. Comp., 65 (1996), 393-396

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)

Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 4, 2013.

F. Oort, Prime numbers, 2013.

Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.

Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.

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

Vmoraru, PlanetMath.org, Germain prime

Eric Weisstein's World of Mathematics, Sophie Germain Prime

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Wikipedia, Sophie Germain prime

MAPLE

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

MATHEMATICA

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

PROG

(MAGMA) [ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

(PARI) select(p->isprime(2*p+1), primes(1000)) \\ Newer versions

select(primes(1000), p->isprime(2*p+1)) \\ v. 2.4.2 and older

(PARI) forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

CROSSREFS

Cf. A005385, A007700, A023272, A023302, A023330, A057331, A005602, A087634.

Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212.

Sequence in context: A131101 A188534 A214196 * A118571 A118504 A215714

Adjacent sequences:  A005381 A005382 A005383 * A005385 A005386 A005387

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Jul 11 1991

STATUS

approved

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Last modified August 22 17:25 EDT 2014. Contains 245975 sequences.