Erdős Problem #415

declared status 'open'. Formalized: no. For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that$$F(n)=(c+o(1))\log\log\log n$$for some constant $c$? Is the first pattern which fails to appear always$$\phi(m+1)>\phi(m+2)>\cdots \phi(m+k)?$$Is it true that 'natural' ordering which mimics what happens to $\phi(1),\ldots,\phi(k)$ is the most likely to appear? Prize: no. Tags: number theory.

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