# 2016-21 Bound on the number of divisors

For a positive integer $$n$$, let $$d(n)$$ be the number of positive divisors of $$n$$. Prove that, for any positive integer $$M$$, there exists a constant $$C>0$$ such that $$d(n) \geq C ( \log n )^M$$ for infinitely many $$n$$.

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# 2011-17 Infinitely many solutions

Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.

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