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In this paper, we consider properties of coefficients of a generatingfunctions composition, where the outer function is a logarithmic generatingfunction and the inner function is an ordinary generating function with integercoefficients. Using notions of composita and composition of generatingfunctions, we get new properties for this composition. The properties can beused for distinguishing prime numbers from composite numbers. As anapplication, obtained results can be used to obtain new primality criteria. Weobtain primality criteria for the Mersenne numbers, the Lucas numbers, thePell-Lucas numbers, the Jacobsthal-Lucas numbers, and the Lucas sequences.Keywords: generating function, composition of generating function, composita,primality criterion.

New properties for a composition of some generating functions for primes