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2^{w(n)} \le k\sqrt[4]{n}. no of (positive) prime divisors related inequality

Source: Ukrainian TST 1999 p10

February 13, 2020
number theoryprime divisorsinequalitiesmin

Problem Statement

For a natural number nn, let w(n)w(n) denote the number of (positive) prime divisors of nn. Find the smallest positive integer kk such that 2w(n)≤kn42^{w(n)} \le k \sqrt[4]{ n} for each n∈Nn \in N.
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