OFFSET
1,1
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.
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).
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Dec 04 2008, Dec 05 2008, Dec 08 2008, Dec 12 2008
STATUS
approved