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Problem of the Day #82: Logarithms of Factors of Factorials June 9, 2011

Posted by Alex in : potd , trackback

Let $f(x) = x!$, $g(x) = \displaystyle\sum_{k | x} \log(k)$, and $h(x) = e^{g(x)}$. A number $x$ is a ffactor of another number $y$ if there exists two integer sequences, $a_i$ and $b_i$ such that $x = \displaystyle\prod_{p_{x_i}} p_{x_i}^{a_i}$ and $y = \displaystyle\prod_{p_{y_i}} (p_{y_i}^{a_i})^{b_i}$ where $p_{x_i}$ is the $i^{\text{th}}$ prime factor of $x$. Find the largest integer value of $x \le 40$ such that $f(x)$ is a ffactor of $h(f(x))$.

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