On a class of solvable recurrences with primes

Mihai Caragiu, Alexandru Zaharescu, Mohamm Zaki

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate an interesting new class of "greatest prime factor sequences" {u n} n≥1 in which every term is the greatest prime factor of the sum of all of the preceding terms. We show that these sequences are explicitly solvable, satisfying a fairly regular growth pattern. Thus, if p n is the nth prime, then the number of occurrences of each large enough p n is p n+1-p n-1 By using a known upper bound for the gaps between consecutive primes, it turns out that the asymptotic estimate u n = n/2 + O(n 0.525) holds true.

Original languageEnglish (US)
Pages (from-to)197-208
Number of pages12
JournalJP Journal of Algebra, Number Theory and Applications
Volume26
Issue number2
StatePublished - Sep 1 2012
Externally publishedYes

Keywords

  • Greatest prime factor
  • Prime gaps
  • Recurrent sequences

ASJC Scopus subject areas

  • Algebra and Number Theory

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