Abstract: How much dependence is there in the prime factorization of a random integer distributed uniformly from 1 to $n$? We show that for primes, $2+\epsilon$ changes -- one deletion and a random number of insertions -- are necessary and sufficient to convert a uniformly distributed random integer from 1 to $n$ into a random integer $\prod_{p \leq n} p^{Z_p} $ in which the multiplicity $Z_p$ of the factor $p$ is geometrically distributed, with all $Z_p$ independent.
Our proof that $2+\epsilon$ suffices uses a coupling of the infinite independent model of prime multiplicities, with the scale invariant Poisson process on $(0,\infty)$. A corollary of this construction is a bound on the distance to the Poisson-Dirichlet in Billingsley's 1972 weak convergence result, of the form: there are couplings in which $$ \e \sum |\log P_i(n) - (\log n) V_i | = O( \log \log n), $$ where $P_i$ denotes the $i^{th}$ largest prime factor and $V_i$ denotes the $i^{th}$ component of the Poisson-Dirichlet process. It is reasonable to conjecture that $O(1)$ is achievable.