On strings of consecutive integers with a distinct number of prime factors

Abstract

Let ω ( n ) \omega (n) be the number of distinct prime factors of n n . For any positive integer k k let n = n k n=n_k be the smallest positive integer such that ω ( n + 1 ) , … , ω ( n + k ) \omega (n+1),\ldots ,\omega (n+k) are mutually distinct. In this paper, we give upper and lower bounds for n k n_k . We study the same quantity when ω ( n ) \omega (n) is replaced by Ω ( n ) \Omega (n) , the total number of prime factors of n n counted with repetitions.