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Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence?

Consider the sum of the prime counting function.

$$ a(n)=\sum_{k=1}^{n} \pi(k) $$

Let $S(n)$ be a string of length $n$, then $a(n)$ is the number of substrings of $S(n)$ with a prime number of characters. Example $1$: "$abcd$" is a string of length $4$; there are $a(4)=5$ substrings with a prime number of characters $(ab, bc, cd, abc, bcd)$. Example $2$: "$abcde$" is a string of length $5$; there are $a(5)=8$ substrings with a prime number of characters $(ab, bc, cd, de, abc, bcd, cde, abcde)$.

What I'm asking is, when is the number of substrings of $S(n)$ with a prime number of characters itself prime?

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  • $\begingroup$ Do you have any idea on where to start ? Have you tried to look at the values such that this property is realized at least in the first 100 integers ? $\endgroup$ – Gâteau-Gallois Apr 20 at 20:24
  • $\begingroup$ As a start one might consider the asymptotic growth of $f(x)=\sum\limits_{n=1}^x b(n)$ where $b(n)=\begin{array}{cc} \{ & \begin{array}{cc} 1 & a(n)\in \mathbb{P} \\ 0 & \text{True} \\ \end{array} \\ \end{array}$ and $a(n)=\sum\limits_{k=1}^n\pi(k)$. For small ranges of $x$, $f(x)$ seems to evaluate fairly closely to $\frac{x}{\log(x)}$. $\endgroup$ – Steven Clark Apr 20 at 22:13

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