Let α(x) : √ x 2 , √ 2 3 , √ 3 4 , √ 4 5 , ... be a sequence with a real variable x > 0 and let Wα(x) be the associated weighted shift with weight sequence α(x). In [17], Exner-Jung-Park provided an algorithm to distinguish weak k-hyponormality and k-hyponormality of weighted shift Wα(x), and obtained sn > 0 for some low numbers n = 4, ..., 10, such that Wα(sn) is weakly n-hyponormal but not n-hyponormal. In this paper, we obtain a formula of sn (for all positive integer n) such that Wα(sn) is weakly n-hyponormal but not n-hyponormal, which improves Exner-Jung-Park’s result above