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Problem 920

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Tau Numbers

For a positive integer $n$ we define $\tau(n)$ to be the count of the divisors of $n$. For example, the divisors of $12$ are $\{1,2,3,4,6,12\}$ and so $\tau(12) = 6$.

A positive integer $n$ is a tau number if it is divisible by $\tau(n)$. For example $\tau(12)=6$ and $6$ divides $12$ so $12$ is a tau number.

Let $m(k)$ be the smallest tau number $x$ such that $\tau(x) = k$. For example, $m(8) = 24$, $m(12)=60$ and $m(16)=384$.

Further define $M(n)$ to be the sum of all $m(k)$ whose values do not exceed $10^n$. You are given $M(3) = 3189$.

Find $M(16)$.

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陶数

对于正整数$n$，定义$\tau(n)$为$n$的约数个数。例如，$12$的约数为$\{1,2,3,4,6,12\}$，因此$\tau(12) = 6$。

如果一个正整数$n$能被$\tau(n)$整除，则称之为陶数。例如，$\tau(12)=6$，且$6$能整除$12$，所以$12$是一个陶数。

记$m(k)$为满足$\tau(x) = k$的最小陶数$x$。例如，$m(8) = 24$，$m(12)=60$，$m(16)=384$。

另记$M(n)$为所有不超过$10^n$的$m(k)$之和。已知$M(3) = 3189$。

求$M(16)$。

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