The distribution of the number of factors in a factorization
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A factorization of a positive integer n, here, is a specification of m(d), the power to which d occurs in dm(d) = n; order is immaterial. The number of factors in a factorization has two natural interpretations: as m(d) or as the number of nonzero m(d), that is, counting or not counting multiplicity. In either case, the factorizations of positive integers x into k factors number approximately v(u) (log x)k - 1 k!(k - 1)!, where u = k(k - 1) log x, and v is either (2 - u) or 1 (1 + u) according to whether multiplicity is counted or not. In the former case, we must have u 2 - ; in the latter, u C. 1987.