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Consider a number $n$ with prime factorization $n=p_1^{k_1} \cdot p_2^{k_2} \dots \cdot p_z^{k_z}$. We define a function $f(n)$ to be $f(n)=(p_1^{k_1+1}-1) \cdot (p_2 ^{k_2 +1}-1) \dots \cdot (p_z^{k_z+1}-1)$. Let $S$ be the set that contains all odd integers $n > 2$ such that $n|f(n)$. What properties of $S$ can be deduced? More specifically:

Is $S$ finite or infinite? What are some numbers that can be part of $S$? Is there a general form to these numbers?

I really dont know where to start, at all. Thanks for the help!

  • 3
    The first member of $S$ seems to be $819 = (3^2)(7)(13)$, which has $f(819) = (256)(819)$. There are no others up to $10^6$.2012-01-28
  • 6
    Echoing Will Jagy's question: why? Is there some reason to be interested in this set?2012-01-28

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