Question 47
a) For each $k \ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ positive divisors of $n$ that are different from each other.
b) Suppose that $n$ can be expressed as the sum of exactly $k$ positive divisors of $n$ distinct from each other, for some $k \ge 3$. Let $p$ be the smallest prime divisor of $n$.
show that
$$
\frac{1}{p} + \frac{1}{p+1} + \dots + \frac{1}{p+k-1} \ge 1.
$$
b) Suppose that $n$ can be expressed as the sum of exactly $k$ positive divisors of $n$ distinct from each other, for some $k \ge 3$. Let $p$ be the smallest prime divisor of $n$.
show that
$$
\frac{1}{p} + \frac{1}{p+1} + \dots + \frac{1}{p+k-1} \ge 1.
$$