Question 134
Prove that there is a positive real number $c\gt 0$ such that there are infinitely many pairs $(n,N)$ of positive integers satisfying the following conditions:
(a) $N\gt c\cdot2^k n\log n$, where $k$ is the number of distinct prime factors of $N$;
(b) $n$ is divisible by $\varphi(p^e)$ for every prime power $p^e$ exactly dividing $N$, where $\varphi$ denotes Euler's totient function: $\varphi(s)$ is the number of positive integers that do not exceed $s$ but are coprime to $s$.
(a) $N\gt c\cdot2^k n\log n$, where $k$ is the number of distinct prime factors of $N$;
(b) $n$ is divisible by $\varphi(p^e)$ for every prime power $p^e$ exactly dividing $N$, where $\varphi$ denotes Euler's totient function: $\varphi(s)$ is the number of positive integers that do not exceed $s$ but are coprime to $s$.