A random integer N, drawn uniformly from the set (1,2,…,n), has a prime factorization of the form N = α1 α2…αM where α1 ⩾ α2 ⩾ … ⩾ αM. We establish the asymptotic distribution, as n → ∞, of the vector A(n) = (log α1,/log N: i: ⩾ 1) in a transparent manner. By randomly re-ordering the components of A(n), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite-dimensional simplex of vectors (x1,x2,…) having non-negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(n) to the corresponding Poisson–Dirichlet distribution on this simplex; this result was obtained by Billingsley [3].
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Donnelly et al. (1993) studied this question.