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On a Class of Solvable Recurrences with Primes (journal paper)

Mihai Caragiu, Alexandru Zaharescu (UIUC), and Mohammad Zaki
Ohio Northern University

Abstract: We investigate an interesting new class of “greatest prime factor sequences” (u_n)_{n\ge 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 n-th 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.

JP Journal of Algebra, Number Theory and Applications Volume 26, Issue 2, Pages 197 - 208 (September 2012)