A156284 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.

3, 7, 11, 17, 19, 23, 31, 37, 43, 59, 67, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 151, 157, 163, 179, 181, 191, 193, 199, 211, 223, 227, 229, 241, 251, 257, 263, 269

Offset: 1

Author

Vladimir Shevelev, Feb 07 2009

Keywords

• nonn

Comments

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).

Crossrefs

Cf. A002375, A152451, A156537.