A152451 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.

3, 7, 17, 23, 31, 37, 43, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 151, 157, 163, 179, 181, 191, 193, 199, 211, 223, 229, 241, 257, 263, 269, 277, 281, 293, 307, 311, 317, 337, 347, 353, 359, 367, 379, 383, 389, 397, 401, 419, 421, 431, 443

Offset: 1

Author

Vladimir Shevelev, Dec 04 2008, Dec 05 2008, Dec 08 2008, Dec 12 2008

Keywords

• nonn

Comments

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

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.

Crossrefs

Complement of A086081.

Cf. A152522.

Programs

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

Formula

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).