Finding a solutions for an equation

nt.quantity idea – On the determinant $det[gcd(i-j,n)]_{1le i,jle n}$ retort

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nt.quantity idea – On the determinant $det[gcd(i-j,n)]_{1le i,jle n}$

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

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

I endure that $D_n$ is all the time optimistic. How to show this?

It is unostentatious to graze that $D_n$ is divisible by $sum_{ok=1}^ngcd(ok,n)=sum_{dmid n}varphi(d)frac nd$.
It appears that $varphi(n)^{varphi(n)}sum_{ok=1}^ngcd(ok,n)$ divides $D_n$. Maybe there’s a unostentatious rationalization for this.

Your feedback are welcome!

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