A subgroup of index p of a finite p-group G is called a k-maximal subgroup of G. Denote by d( G) the number of elements in a minimal generator-system of G and by δ( G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δ( G) ≤ p and δ−1( G) = 0, respectively.