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Abstract. Rényi’s result on the density of integers whose prime factorizations have excess multiplicity has an analogue for polynomials over a finite field. Let n = pα11 · · · pαrr be the prime factorization of a positive integer n. Define the excess of n to be (α1−1)+ · · ·+(αr−1), which is the difference between the total multiplicity α1 + · · · + αr and the number of distinct primes in the factorization. An integer with excess 0 is also said to be square-free. Let Ek denote the set of positive integers of excess k, k = 0, 1, 2,.... Rényi proved that the set Ek has a density dk and that the sequence {dk} has a generating function given by∑ k≥0 dkz k