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Representations of k as product of integers is bounded

Source: 2014 China TST 3 Day 2 Q6

April 5, 2014
inductionnumber theory proposednumber theory

Problem Statement

For positive integer k>1k>1, let f(k)f(k) be the number of ways of factoring kk into product of positive integers greater than 11 (The order of factors are not countered, for example f(12)=4f(12)=4, as 1212 can be factored in these 44 ways: 12,2⋅6,3⋅4,2⋅2⋅312,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3. Prove: If nn is a positive integer greater than 11, pp is a prime factor of nn, then f(n)≤npf(n)\leq \frac{n}{p}
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