A092816 - OEIS

%I #74 Aug 05 2023 11:23:25

%S 3,10,37,190,1171,7746,56032,423140,3308859,26569515,218116524,

%T 1822848478,15462601989,132822315652

%N Number of Sophie Germain primes less than 10^n.

%C 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

%D P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.

%H C. K. Caldwell, <a href="http://www.utm.edu/~caldwell/preprints/Heuristics.pdf">An amazing prime heuristic</a>, Table 6.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain Prime</a>

%F 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]

%F 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

%e 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.

%Y Cf. A005384, A156874, A182265.

%K nonn,more

%O 1,1

%A _Eric W. Weisstein_, Mar 06 2004

%E a(10) computed by _Eric W. Weisstein_, Nov 02 2005

%E a(11)-a(12) from _Donovan Johnson_, Jun 19 2010

%E a(13)-a(14) from _Giovanni Resta_, Sep 04 2017