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We consider the recursion sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where ${p_i}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..

The first few values of $R_n$ are: 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, ...

Now, we define $R(n)$ to be the number of prime numbers in the set ${R_0, R_1 , \dots , R_n}$.

What I have found (without justification) is that $R(n) \approx \frac{2n}{\ln (n)}$.

My lack of programming skills, however, prevents me from checking further numerical examples. I was wondering if anyone here had any ideas as to how to prove this assertion.

As a parting statement, I bring up a quote from Gauss, which I feel describes many conjectures regarding prime numbers: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."

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We consider the recursion $R_n=p_n+p_{n+1}+p_{n+2}$, where ${p_i}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..

The first few values of $R_n$ for $n=0,1,2,\dots$ are: $10, 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199,$ $211,223,235,251,269,287,301,311,319,329,349,271,395,407,425,439, 457$

$\dots \dots \dots$211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, ...

Now, we define $R(n)$ to be the number of prime numbers in the set ${R_0, R_1 , \dots , R_n}$.[b]

What I have found (without justification) is that $R(n) \approx \frac{2n}{\ln (n)}$[/b]n)}\$.

My lack of programming skills, however, prevents me from checking further numerical examples. I was wondering if anyone here had any ideas as to how to prove this assertion.

As a parting statement, I bring up a quote from Gauss, which I feel describes many conjectures regarding prime numbers: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."