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Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) showed that $\rho^\ast(x)>\pi(x)$ for $x=20000$, hence under the prime tuples conjecture we also have $\pi(x+y)>\pi(x)+\pi(y)$ for the same $x$ and some $y$.

Clark and Jarvis (2001) found that already $x=4916$ satisfies $\rho^\ast(x)>\pi(x)$. Are their any smaller values of $x$ available, has anyone worked on this question since? Clark and Jarvis (2001) also showed that $\rho^\ast(x)\leq \pi(x)$ for $2\leq x\leq 1426$.

Added. As Wojowu remarked, $x=3159$ is also known to satisfy $\rho^\ast(x)>\pi(x)$.

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    $\begingroup$ Wikipedia tells us that $3159$ works. $\endgroup$
    – Wojowu
    Commented Sep 13, 2017 at 16:01
  • $\begingroup$ @Wojowu: Thank you! This Wikipedia entry is a good reference. $\endgroup$
    – GH from MO
    Commented Sep 13, 2017 at 16:02
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    $\begingroup$ This page seems to analyze things in great detail. In particular, it tells us that the smallest such $x$ satisfies $2529<x\leq 3159$. Last update was in 2009 though, so progress might've been made. $\endgroup$
    – Wojowu
    Commented Sep 13, 2017 at 16:04
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    $\begingroup$ @Wojowu: Thank you again. I recommend that you write up your comments as a response. (I will wait a few days and then accept the best response up to that point.) $\endgroup$
    – GH from MO
    Commented Sep 13, 2017 at 16:07

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