In Sept. 2013, I investigated the determinant $$D_n=\det[\gcd(i-j,n)]_{1\le i,j\le n}$$ and computed the values $D_1,\ldots,D_{100}$ (cf. http://oeis.org/A228884). To my surprise, they are all positive!

**Question.** Does $D_n>0$ hold for all $n=1,2,3,\ldots$?

I believe that $D_n$ is always positive. How to prove this?

It is easy to see that $D_n$ is divisible by $\sum_{k=1}^n\gcd(k,n)=\sum_{d\mid n}\varphi(d)\frac nd$. It seems that $\varphi(n)^{\varphi(n)}\sum_{k=1}^n\gcd(k,n)$ divides $D_n$. Maybe there is a simple explanation for this.

Your comments are welcome!