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I have the following question:

Is there a closed formula for the elementary divisors of the Matrix $M={(m_{ij})}_{i=1,...,n,\ j=1,...,k}$, where ${m}_{ij}$ is the greates common divisor of $i$ and $j$?

I know that det$(M)=\varphi(1)\cdot ... \cdot \varphi(n)$, if $M$ is a$\ $ $n\times n$ square matrix.

Here, $\varphi$ is Euler's totient function.

But how to compute the elementary divisors? Is it easy to compute the determinant divisors of $M$?

Thank you very much.

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