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\begin{document}

$\;$\hfill Due: 2012/10/17 before class

\begin{center}
{\LARGE\bf Homework 4}
\end{center}

\begin{hw}
Let $n$ be a positive integer, determine $\sum_{1 \leq m \leq n} \mu(m) \lfloor n / m \rfloor$. Justify your answer.
\end{hw}

\begin{hw}
Let $P(k)$ be the number of distinct prime factors of $k$. e.g., $P(1) = 0$, $P(2)=P(8)=P(25)=1$, and $P(30)=3$. Calculate
\[
\sum_{d | 20121009} \mu(d) P(\frac{20121009}{d})
\]
and justify your answer.
\end{hw}

\begin{hw} Let f(n) be the number of integer partitions of $n$ into
$x_1 + x_2 + ... + x_t$, where $t$ is any number and $x_i = x_{i+1} + 2$ for all
$1 \leq i < t$. Prove that $f(n)$ equals the number of divisors of
  $n$ that are at most $\sqrt{n}$.
\end{hw}

\begin{hw}
(a). Compute $p(n)$  by writing out all the unordered partitions of $n$,
for $n = 1, 2, 3, 4, 5$.

(b). Prove $p(n) \leq F_n$, where $F_n$ is the $n$-th Fibonacci number.
\end{hw}

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