A generalized repunit probable prime (base -373).
24007 is the smallest odd prime p such that (373^p+1)/374 is a (probable) prime.
For the smallest odd prime p such that (n^p+1)/(n+1) is a (probable) prime (2<=n<=512, n is not a perfect odd power or of the form 4*k^4), the top 10 currently known (n,p) are (sorted by p): (365,25579), (373,24007), (188,22037), (53,21943), (124,16427), (284,9203), (356,7937), (340,7001), (150,6883), (318,6733).
Currently, there are no known (probable) primes of the form (n^p+1)/(n+1) with odd prime p for n = 97, 103, 113, 175, 186, 187, 220, 304, 306, 309, 316, 335, 414, 416, 428, 433, 434, 445, 459, 486, 498 (less than 512) All of these n have been searched to at least p = 7000 with no primes found.

To be completed...