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Old 2019-12-20, 06:53   #2
miket
 
May 2013

32 Posts
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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 \(4\).
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