Category:Autodesk
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Category:AutodeskQ:

Let $\phi(n)$ be the number of solutions to $xy=n$ in integers $x,y$, if $3\mid n$. $3^{k-1} | \phi(n)$ for all $k$.

Let $\phi(n)$ be the number of solutions to $xy=n$ in integers $x,y$, if $3\mid n$. Is $3^{k-1} | \phi(n)$ for all $k$?
I know that for $k=1$, $3^{k-1} | \phi(n)$ is correct. But I am not sure if $3^{k-1} | \phi(n)$ for all $k$.
I have been trying to find $k$ so that $3^{k-1} | \phi(n)$ for all $k$, but have not been successful.

A:

$\phi(n) = 0$ when $n = 0$ or $n = -3$.
$\phi(n) = 1$ when $n = 1$.
$\phi(n) = 2$ when $n = 4$.
$\phi(n) = 3$ when $n = 9$ or $n = 36$.
$\phi(n) = 4$ when $n = 25$ or $n = 80$.
$\phi(n) = 6$ when $n = 81$ or $n = 216$.
$\phi(n) = 8$ when $n = 324$ or $n = 3600$.
$\phi(n) = 12$ when $n = 4095$ or $n = 259200$.
$\phi(n) = 24$ when $n = 0$ or $n = -3\times3^{11}$ or $n = -3\times3^{14}$.

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