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Conjecture: a(n) > 0 for all n > 2. Also, any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6*z-1 are terms of A230138 and 6*y*z-1 is prime.

This is a further refinement of the conjecture in A230140.

Note that if x + y + z = n then 6*n = (6*x-1) + (6*(y+z)+1). So a(n) > 0 implies Goldbach's conjecture for the even number 6*n.

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 30, then 2*n + 1 can be written as 2*p + q with p, q and prime(p + 2) + 2 all prime.

Part (i) implies both the Goldbach conjecture and the twin prime conjecture. If all primes p with prime(p + 2) + 2 are smaller than an even number N > 2, then for any such a prime p the number N! + N - p is in the interval (N!, N! + N) and hence not prime.

Similarly, part (ii) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture.

We have verified part (i) of the conjecture for n up to 2*10^8.

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).

According to the construction of A156284, a_(3n)=0, n>=1. These terms may be called "wells". The growth of the depth of the "wells" is O(2^(n/3)log(n)/n^2).

Conjecture: (i) a(n) > 0 for all n > 3. Moreover, every n = 6, 7, ... can be written as p + q - pi(q) with p, p + 6 and q all prime.

(ii) For each integer n > 7, there is a prime p < n with n + p - pi(p) prime.

(iii) Any integer n > 4 not equal to 9 or 17 can be written as p + q + pi(q) with p and q both prime.

(iv) Each integer n > 7 can be written as p + q + pi(p) + pi(q) with p and q both prime.

Conjecture: (i) a(n) > 0 for all n > 1.

(ii) For any odd number 2*n - 1 > 4, there is a positive integer k < 2*n such that 2*n - 1 - phi(k) and 2*n - 1 + phi(k) are both prime.

By Goldbach's conjecture, 2*n > 2 could be written as p + q with p and q both prime, and hence 2*n - 1 = p + (q - 1) = p + phi(q).

By induction, phi(k^2) (k = 1,2,3,...) are pairwise distinct.

The Goldbach conjecture, if true, would imply a(n) > 0.

Row lengths of table A260689, n > 1. - Reinhard Zumkeller, Nov 17 2015

The above sequence relies on the strong Goldbach's conjecture that any positive integer is the sum of at most three distinct terms from {1 union primes}.

Conjecture: a(n) > 0 for all n > 784.

This conjecture implies Goldbach's conjecture, Lemoine's conjecture, and that there are infinitely many primes of the form p^2 + q^2 - 1 with p and q both prime.

It has been verified for n up to 10^8.

Zhi-Wei Sun also made the following general conjecture: Let d be any odd integer not congruent to 1 modulo 3. Then, all large even numbers can be written as p + q with p, q, p^2 + q^2 + d all prime. If d is also not divisible by 5, then all large odd numbers can be represented as p + 2q with p, q, p^2 + q^2 + d all prime.

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5. Moreover, if n = 6*k + 2, then a(n) > 0 except for k = 0, 1, 12, 28, 102, 117, 168, 4079.

We have verified the conjecture for n up to 10^9.