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Let $\varphi$ be Euler's totient function. If $p$ is a prime, then $\varphi(1+n)=n$ for $n=p-1$.

Question. Is it true that for each integer $m>1$ there is a positive integer $n\le m^2-m$ such that $\varphi(m+n)\mid n$ ?

I conjecture that the question has a positive answer. See http://oeis.org/A248568 for related data. For example, $\varphi(10+40)=20$ divides $40$.

Your comments are welcome!

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