I'm inspired in the so-called Erdős squarefree conjecture, this section from Wikipedia, to state in this post a question, involving a different arithmetic function, that due its difficulty I feel as an exercise (in the Appendix I show another example), and in my next post I am going to add a different variation that I think that due its difficulty should be a conjecture.
In this post we denote the Euler's totient function as $\varphi(n)$, and it is well-known that satisfies $$\varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)$$ for integers $n>1$.
Also is well-known the multiplicative formula for factorials, this section of Wikipedia.
Computational fact. In the segment of positive integers $1\leq n\leq 20000$, the integer $n=160$ is the last integer for which $$\binom{2n}{\varphi(n)}$$ is a square-free integer (has no repeated prime factors).
The sequence of binomial coefficients $$\binom{2n}{\varphi(n)}$$ having some repeated prime factors (when $n\geq 1$ runs over integers) starts as $$4,28,1820,18564\ldots$$
Question. I belive that the following claim is true:
The binomial coefficient $\binom{2n}{\varphi(n)}$ is never squarefree for $n>160$.
Can you prove it or provide me hints? Many thanks.
Now I am going to add in next appendix a different example that I think more easy than previous, with the purpose to provide it to everyone that is interested in this kind of questions.
Appendix: We (exclude the case $n=2$) consider the arithmetic function $\binom{n\varphi(n)}{\sigma(n)}$, where $\sigma(n)=\sum_{d\mid n}d$ denotes the sum of divisors function. Then I believe that the following claim is true (you can to study it in your home, but feel free to add a counterexample as a comment if you find it).
Claim. The binomial coefficient $$\binom{n\varphi(n)}{\sigma(n)}$$ is never squarefree for $n>22$.