Let \(\varvec{N=\displaystyle \prod _{i=1}^{k}p_i^{a_i}}\) be a fixed positive integer with known prime factorization. We investigate bootstrap resampling from the prime-factor multiset of \(N\) and study two associated statistics, namely the log-product of the resampled integer and the number of distinct observed primes appearing in the resample. It is shown that the log-product has explicit mean and variance determined by the empirical law of the logarithms of the observed primes, while bootstrap- \(\omega \) is an occupancy sum with explicit mean, variance, and negative pairwise covariance structure. In the square-free case, exact formulas are obtained and asymptotic constants are identified, yielding a benchmark on the Erdős–Kac scale in random-like regimes. Numerical experiments on a synthetic corpus of factorizations illustrate the formulas, the square-free benchmark, and the behavior of the diagnostics across smooth, square-free, spiky, and mixed factorization regimes.
