%I #26 Jan 23 2023 09:11:33

%S 3,7,17,23,31,37,43,71,73,79,83,89,101,103,107,113,127,131,137,139,

%T 151,157,163,179,181,191,193,199,211,223,229,241,257,263,269,277,281,

%U 293,307,311,317,337,347,353,359,367,379,383,389,397,401,419,421,431,443

%N From every interval (2^(m-1), 2^m), we remove primes p for which 2^m-p is a prime; the sequence gives the remaining odd primes.

%C Powers of 2 are not expressible as sums of two primes from this sequence.

%C Consider a strong Goldbach conjecture: every even number n >= 6 is a sum of two primes, the lesser of which is O((log(n))^2*log(log(n))) (cf. comment to A152522). The number of such representations for 2^k, trivially, is less than k^5 for k > k_0. Removing the maximal primes in every such representation of 2^k, k >= 3, we obtain an analog B of A152451 with the counting function H(x) = pi(x) - O((log(x))^5). Replacing in B the first N terms with N consecutive primes (with arbitrarily large N), we obtain a sequence which essentially is indistinguishable from the sequence of all primes with the help of the approximation of pi(x) by li(x), since, according to the well-known Littlewood result, the remainder term in the theorem of primes cannot be less than sqrt(x)logloglog(x)/log(x). But for this sequence we have infinitely many even numbers for which the considered strong Goldbach conjecture is wrong. Thus the conjecture is essentially unprovable.

%F If A(X) is the counting function for the terms a(n)<=x, then A(x) = x/log(x) + O(x*log(log(x))/(log(x))^2).

%o (PARI) lista(nn) = {forprime(p=3, nn, m = ceil(log(p)/log(2)); if (!isprime(2^m-p), print1(p, ", ")););} \\ _Michel Marcus_, Sep 12 2015; Jan 22 2023

%Y Complement of A086081.

%Y Cf. A152522.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Dec 04 2008, Dec 05 2008, Dec 08 2008, Dec 12 2008