Sums of reciprocals of divisors, perfect numbers

A perfect number is a positive integer $n$ such that
$$\sum_{d|n} d = 2n.$$Put another way, $n$ is the sum of its proper divisors. Check out a a quick intro to perfect numbers that I wrote last November. The first three perfect numbers are $6, 28,$ and $496$. Currently, the largest perfect number, corresponding to the largest Mersenne prime is
$$(2^{77232917} – 1)\cdot 2^{77232916}$$This perfect number is over 46 million digits long!

Here’s another fun fact about perfect numbers. A positive integer $n$ is perfect if and only if
$$\sum_{d\mid n} \frac{1}{d} = 2.$$This is just a slightly disguised form of the definition of perfect. For
$$\sum_{d\mid n} \frac{1}{d} = \sum_{d\mid n} \frac{n/d}{n} = 2n/2.$$ Even so, the sum $\sum_{d\mid n} 1/d$ is a cool looking as a function of $n$:

What are those stronger lines all about? I don’t know. Is this sum bounded? No, because $\sum_{d\mid n} 1/d$ contains the first $k$ terms of the harmonic series for $n = k!$.

Speaking of sums of divisors, can a number ever be the sum of squares of all its proper divisors? What about cubes? I couldn’t find any examples…can you?

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