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Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$

For all terms of $A$ greater than $\ \varphi(d) \cdot d^2: \ $ each term either is a prime number or is the average of two primes which are terms of $A$.

Are there conjectures similar to this conjecture?

As Lucia commented: The Goldbach problem is similar to this conjecture. What's the precise relationship between these two problems?

$\ \varphi(d) \cdot d^2 \ $ see: http://oeis.org/A053191
Source (in Chinese): http://www.mitbbs.com/article_t/Mathematics/31205723.html

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    $\begingroup$ The Goldbach problem is similar to this conjecture. $\endgroup$ – Lucia Jul 7 '14 at 14:16
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    $\begingroup$ Indeed, this is a Dirichlet version of Goldbach. The conjecture implies Linnik's theorem with an exponent of 3 or better. $\endgroup$ – The Masked Avenger Jul 7 '14 at 17:35
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    $\begingroup$ Also posted to m.se, math.stackexchange.com/questions/858764/… --- come on, Mike, you know better than to post the same question to two different sites without linking them. $\endgroup$ – Gerry Myerson Jul 8 '14 at 0:24

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