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