On a class of solvable recurrences with primes

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.