KöMaL problem A, 2015

Number theory Difficulty 6.0 National Olympiad Prove it Hungary

Show that for every positive real number $c>0$ there is a positive integer $n$ such that $\varphi\big(\sigma(n)\big)>cn$ . (For an arbitrary postive integer $k$ , $\varphi(k)$ denotes the number of positive integers not exceeding $k$ that are co-prime with $k$ . $\sigma(k)$ is the sum of positive divisors of $k$ .)
Proposed by: Barnabás Szabó, Budapest
(5 pont)

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