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 2019-12-20, 06:53 #2 miket   May 2013 32 Posts Robert Israel answer this question at math.stackexchange Does the constant 4.018 exists?: What you mean is, if $$b_n$$ is the $$n$$'th nonnegative integer $$j$$ such that $$3j^2+2$$ is prime, $\lim_{n \to \infty} \dfrac{\sum_{j=1}^n (3 b_j^2+2)}{\sum_{j=1}^n (b_j^3-b_j)} b_n \approx 4.018$ We don't even know for certain that there are infinitely many such integers, but heuristically it is likely that $$b_j \sim c j \log j$$ for some constant $$c$$ as $$j \to \infty$$. If so, $$\sum_{j=1}^n (3 b_j^2 + 2) \sim c^2 n^3 \log^2 n$$, $$\sum_{j=1}^n (b_j^3 - b_j) \sim c^3 n^4 \log^3(n)/4$$, and so your limit should be exactly $$4$$.