ｎ为自然数（ｎ＞１）令ｐ（ｎ）表示ｎ的最小素因子，（ｎ）表示。的所有的素因子的个数，ｗ（ｎ）表示ｎ的不同的素因子的个数．本文绘出了，的渐近估计式，其中ｒ＞０．它们改进并推广了张文鹏相应的结果．

For the natural number n > 1, let p(n) denote the smallest prime factor ofn, (n) denote the total number of the prime faCtors of n, w(n) denote the number ofthe different prime factors of n. This paper produces the estimates for the followingsums, such as which improves the results of zhang Weipeng.