The least primary factor of the multiplicative group

Let [Formula: see text] denote the least primary factor in the primary decomposition of the multiplicative group [Formula: see text]. We give an asymptotic formula, with order of magnitude [Formula: see text], for the counting function of those integers n for which [Formula: see text]. We also give an asymptotic formula, for any prime power q, for the counting function of those integers n for which [Formula: see text]. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.