1,1

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).

Jacob Korevaar, Lower bound for the remainder in the prime-pair conjecture

a(n) = #{p < 10^n in A028870}.

a(1) = 4 because {2, 3, 5, 7} are the 4 primes p less than 10^1 such that p^-2 are primes, namely {2, 7, 23, 47}.

a(2) = 13 = #{2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89}.

Cf. A000040, A028870.

Sequence in context: A149464 A129147 A151476 this_sequence A149465 A149466 A006604

Adjacent sequences: A140451 A140452 A140453 this_sequence A140455 A140456 A140457

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 26 2008