Using formulae for one- and two-electron integrals of Coulomb interaction potential f(k)(r) = r(-k) with non-integer indices k established by one of the authors with the help of complete orthonormal sets of Psi(alpha)-exponential-type orbitals (alpha = 1, 0, -1, -2, . . .), we perform the calculations for isoelectronic series of the He atom containing nuclear charges from 2 to 10, where k = 1 - mu (-1 < mu < 0). For this purpose we have used the double-zeta approximation, the configuration interaction and coupled-cluster methods employing the integer-n Slater-type orbitals as basis sets. It is demonstrated that the results of calculations obtained are better than the numerical Hartree-Fock values.