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Let ω(n) denote the number of prime divisors of n and let Ω(n) denote the number of prime power divisors of n. We obtain upper bounds for the lengths of the longest intervals below x where ω(n), respectively Ω(n), remains constant. Similarly we consider the corresponding problems where the numbers ω(n), respectively Ω(n), are required to be all different on an interval. We show that the number of...
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