ON ANOTHER ERDÖS’ OPEN PROBLEM

Paul Erdös has proposed the following problem: (1) “Is it true that lim n max m n (m d(m)) n?, where d(m) represents the number of all positive divisors of m.” We clearly have: Lemma 1. * 1 ( ) \ 0,1,2, ()! , ()!,..., , 0s sn sN N N, such that n p1 1 ps s 1, where p1, p2,... constitute the increasing sequence of all positive primes. Lemma 2. Let s *. We define the subsequence ns (i) p1 1 ps s 1, where 1,..., s are arbitrary elements of, such that s 0 and 1... s and we order it such that ns(1) ns(2)... (increasing sequence). We find an infinite number of subsequences ()sn i, when s traverses * , with the properties: a) lim i ns (i) for all s b