On locally repeated values of certain arithmetic functions, IV

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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 solutions g (n) to the equation m + ω(m) = n is an unbounded function of n, thus answering a question posed in an earlier paper in this series. A principal tool is a Turán-Kubilius type inequality for additive functions on arithmetic progressions with a large modulus.

Original languageEnglish
Pages (from-to)227-241
Number of pages15
JournalRamanujan Journal
Issue number3
Publication statusPublished - Jan 1 1997


  • Additive functions
  • Prime divisors
  • Turán-kubilius inequality

ASJC Scopus subject areas

  • Algebra and Number Theory

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