OFFSET
1,1
COMMENTS
Hardy-Littlewood conjecture: Number of Sophie Germain primes less than n ~ 2*C2*n/(log(n))^2, where C2 = 0.6601618158... is the twin prime constant (see A005597). The truth of the above conjecture would imply that there are an infinite number of Sophie Germain primes (which is also conjectured). - Robert G. Wilson v, Jan 31 2013
REFERENCES
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.
LINKS
C. K. Caldwell, An amazing prime heuristic, Table 6.
Eric Weisstein's World of Mathematics, Sophie Germain Prime
FORMULA
For 1 < n < 15, a(n) ~ e * (pi(2*10^n) - pi(10^n)) / (5*n - 5) where e is Napier's constant, see A001113 (we use n > 1 to avoid division by zero; whether the formula holds for any n > 14 is unknown). - Sergey Pavlov, Apr 07 2021 [This formula fails under the Hardy-Littlewood conjecture; the leading constant is wrong. - Charles R Greathouse IV, Aug 03 2023]
For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(2 * 10^n + 1) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that both c and c' = 2*c + 1 are composite (trivial). - Sergey Pavlov, Apr 08 2021
EXAMPLE
The Sophie Germain primes up to 10 are 2 (since 5 is prime), 3 (since 7 is prime), and 5 (since 11 is prime), so a(1) = 3.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Eric W. Weisstein, Mar 06 2004
EXTENSIONS
a(10) computed by Eric W. Weisstein, Nov 02 2005
a(11)-a(12) from Donovan Johnson, Jun 19 2010
a(13)-a(14) from Giovanni Resta, Sep 04 2017
STATUS
approved