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The transition temperature, the depletion of the condensate atoms and the collective excitations of a Bose-Einstein condensation (BEC) with two- and three-body interactions in an anharmonic trap at finite temperature are studied in detail. By using the Popov version of the Hartree-Fock-Bogoliubov (HFB) approximation, an extended self-consistent model describing BEC with both two- and three-body interactions in a distorted harmonic potential at finite temperature is obtained and solved numerically. The results show that the transition temperature, the condensed atom number and the collective excitations are modified dramatically by the atomic three-body interactions and the distortion of the harmonic trap.