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A092816 Number of Sophie Germain primes less than 10^n.

%I A092816 #75 Feb 16 2025 08:32:53
%S A092816 3,10,37,190,1171,7746,56032,423140,3308859,26569515,218116524,
%T A092816 1822848478,15462601989,132822315652
%N A092816 Number of Sophie Germain primes less than 10^n.
%C A092816 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 A092816 P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.
%H A092816 C. K. Caldwell, <a href="http://www.utm.edu/~caldwell/preprints/Heuristics.pdf">An amazing prime heuristic</a>, Table 6.
%H A092816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain Prime</a>
%F A092816 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 A092816 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 A092816 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 A092816 Cf. A005384, A156874, A182265.
%K A092816 nonn,more
%O A092816 1,1
%A A092816 _Eric W. Weisstein_, Mar 06 2004
%E A092816 a(10) computed by _Eric W. Weisstein_, Nov 02 2005
%E A092816 a(11)-a(12) from _Donovan Johnson_, Jun 19 2010
%E A092816 a(13)-a(14) from _Giovanni Resta_, Sep 04 2017