In order to find the rules of the representation for integers under the factorial base, a kind of digital sum function and its characteristics are studied. Let w (m) denote the digital sum of integer m m(0≤m≤n!-1) in the factorial base. For any positive integer x and any given integer k≥0, a sharp calculating formula of the k - th power of this function is obtained by a mathematical combination method. These results are of perspective value in coding, cryptography and computation complexity theory.