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Old 2019-12-20, 03:19   #1
miket
 
May 2013

916 Posts
Default Does the constant 4.018 exists?

Let \(A\) be the set \(\{a_1,a_2,\ldots,a_n\},\) for each \(i, a_i \)is prime number of the form \(3j^2+2, j \geq 0 \)

let \(B\) be the set \(\{b_1,b_2,\ldots,b_n\}\), for each \(i, 3b_i^2+2\) is prime number,\( b_i \geq 0 \)

Let \[ f(n)=\frac{\quad\sum A}{\quad\sum_{b\in B} b^3 - b}b_n, b \in B\]

For example, when \(n=3\), \[f(3)= \frac{2+5+29}{0^3 - 0 + 1^3 - 1 + 3^3 - 3} \times 3 = 4.5 \]

When \(n=40400\), \[f(40400)=\dfrac{38237010330695965}{9515800255043913608016} \times 999967 \approx 4.018 \]

When \(n=2988619\), \[f(2988619)=\dfrac{28727312822972002780844}{714881028260333643707250890088} \times 99999987 \approx 4.018 \]

Is it possible that
\[\lim_{n\to+\infty}f(n) \approx 4.018?\]

I only check \(b_n\) to \(10^8\), furthermore check are welcome.
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