A138181 Largest Fibonacci number not exceeding the n-th prime.

2, 3, 5, 5, 11, 13, 13, 13, 21, 21, 21, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233