Question

Simplify (a to the power 3n-9 whole to the power 6/a to the power 2n -4

Answer 1

What is the sum of the amount powers of primes p between n−2p and n, with n>p, and2<p<n2?

By powers of primes, I mean an integer of the form pk, not an integer of the form paqb⋯. So far, I have tried this, by letting ψ represent the quantity above:

ψ=∑q(logq(n)−logq(n−2q))=∑qln(nn−2q)ln(q),

where q ranges across the odd primes less than or equal to n2. I'm not sure how to simplify further, or whether the sum above ever gets very large. In case my sum is correct and you see some sort of simplification, please also discuss behavior of ψ as n→∞.