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%I A140454 #9 Jul 06 2021 16:05:18
%S A140454 4,13,52,259,1595,10548,74914,563533,4387106
%N A140454 Number of primes p less than 10^n such that p^2-2 is prime.
%C A140454 Korevaar gives these values in Table 1, p. 18, attributing the calculation to Fokko van de Bult. Abstract: For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive constant C_{2r}. A heuristic argument indicates that the remainder e_{2r}(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_{2r}(x) similar to one of Riemann for pi(x).
%H A140454 Jacob Korevaar, <a href="http://arxiv.org/abs/0806.4057">Lower bound for the remainder in the prime-pair conjecture</a>, arXiv:0806.4057
%F A140454 a(n) = #{p < 10^n in A028870}.
%e A140454 a(1) = 4 because {2, 3, 5, 7} are the 4 primes p less than 10^1 such that p^2-2 are primes, namely {2, 7, 23, 47}.
%e A140454 a(2) = 13 = #{2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89}.
%Y A140454 Cf. A000040, A028870.
%K A140454 nonn
%O A140454 1,1
%A A140454 _Jonathan Vos Post_, Jun 26 2008
%E A140454 a(9) from _Donovan Johnson_, Feb 17 2010

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