Smart observation. Another way to say it is that, for distinct primes p1 and p2, the events “p1 divides n”, and “p2 divides n”, are approximately statistically

Smart observation. Another way to say it is that, for distinct primes p1 and p2, the events “p1 divides n”, and “p2 divides n”, are approximately statistically independent. So you get a near-diagonal covariance with entries as you wrote.