I prefer to make a distinction between recurrence, identity, and formula. Otherwise the whole discussion is moot. For example, to compute the nth fibonacci, we have Binet’s formula. We don’t need to bootstrap - we just take the golden ratio and its conjugate, exponentiate, take the difference and scale by root 5. That’s a genuine formula. otoh If one says the nth fibonacci is just the sum of the previous two fibonaccis, that’s technically not a formula - you would need to bootstrap to get those previous ones, so that a recurrence - still very useful, but not a formula in the sense you can’t turn the crank and be done. In that specific sense, I agree with Wilf and Dudley that the procedures outlined by Mills, Wright, Wilson etc are essentially worthless as formulas. They are nice identities and recurrences, but not a genuine formula in that you can’t get the nth prime by turning the crank - you have to bootstrap with so many extraneous params it makes the whole exercise futile.