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Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function.

As Arithmetic function - Wikipedia mentioned, there is an equation that $$\sigma_k(u) \sigma_k(v) = \sum_{d~| \gcd(u, v)}{d^k \sigma_k\left(\frac{u v}{d^2}\right)}$$ which can also be represented as $$\sigma_k(u v) = \sum_{d~| \gcd(u, v)}{\mu(d) d^k \sigma_k\left(\frac{u}{d}\right) \sigma_k\left(\frac{v}{d}\right)}.$$

I wonder to know if there exists an equation between $\sigma_k(u v w)$ and $\sigma_k(u) \sigma_k(v) \sigma_k(w)$?

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    $\begingroup$ Iteratively apply same formulae. E.g., multiply the first one by $\sigma_k(w)$ and further combine $\sigma_k(uv/d)\sigma_k(w)$ using the same formula. $\endgroup$ – Max Alekseyev Mar 7 '18 at 13:30

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