In this paper the integer valued ω-index theory parameterized by all ω on the unit circle for paths in the symplectic group Sp(2n) is established. Based on this index theory, the Bott formula of the Malsov-type index theory for iterated paths in Sp(2n) is estalished, the mean index for periodic solutions of Hamiltonian systems is defined, and the increasing estimate of the iterated Maslov-type index in terms of the mean index is proved.