Brocard's conjecture states that

for , where is the prime counting function and is the th prime. For , 2, ..., the first few values are 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, ... (OEIS A050216).

More things to try:

Weisstein, Eric W. "Brocard's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardsConjecture.html