Elma, ErtanLiu, Yu-Ru2023-10-032023-10-032022-03https://doi.org/10.4153/s0008439521000266http://hdl.handle.net/10012/20007This article has been published in a revised form in the Canadian Mathematical Bulletin https://doi.org/10.4153/S0008439521000266. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Canadian Mathematical Society 2021Let k ⩾ 1 be a natural number and ωk (n) denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions ωk with k ⩾ 1. Moreover, we prove that the function ω1(n) has normal order log log n and the function (ω1(n) − log log n)/√log log n has a normal distribution. Finally, we prove that the functions ωk (n) with k ⩾ 2 do not have normal order F(n) for any nondecreasing nonnegative function F.enprime divisorsnormal ordernormal distributionArticle