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Sum of highest prime power divisors

Source: Miklós Schweitzer 2018 P5

November 10, 2018
college contests

Problem Statement

For every positive integer nn, define f(n)=∑p∣npkp,f(n)=\sum_{p\mid n}{p^{k_p}},where the sum is taken over all positive prime divisors pp of nn, and kpk_p is the unique integer satisfying pkp⩽n<pkp+1.p^{k_p}\leqslant n<p^{k_p+1}.Findlim sup⁡n→∞f(n)log⁡log⁡nnlog⁡n.\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .
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