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%I #15 Jul 08 2018 07:57:31
%S 3,7,11,17,19,23,31,37,43,59,67,71,73,79,83,89,101,103,107,113,127,
%T 131,137,139,151,157,163,179,181,191,193,199,211,223,227,229,241,251,
%U 257,263,269
%N From every interval (2^(m-1), 2^m), m >= 3, we remove primes p for which 2^m-p is a prime that was not removed for smaller values of m; the sequence gives all remaining odd primes.
%C Powers of 2 are not expressible as sums of two primes from this sequence. This is attained by a more economical algorithm than that for construction of A152451. If A(x) is the counting function for the terms a(n) <= x, then A(x) = pi(x) - O(x/(log^2(x)). It is known that the approximation of pi(x) by x/log(x) gives the remainder term as, at best, O(x/log^2(x)). Therefore beginning our process from m >= M (with arbitrarily large M), we obtain a sequence which essentially is indistinguishable from the sequence of all odd primes with the help of the approximation of pi(x) by x/log(x). Hence it is in principle impossible to prove the binary Goldbach conjecture by such an approximation of pi(x).
%Y Cf. A002375, A152451, A156537.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Feb 07 2009
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