A generalized repunit probable prime (base 306).
26407 is the smallest odd prime p such that (306^p-1)/305 is a (probable) prime.
For the smallest odd prime p such that (n^p-1)/(n-1) is a (probable) prime (2<=n<=1024, n is not a perfect power), the top 10 currently known (n,p) are (sorted by p): (152,270217), (18,25667), (306,26407), (331,25033), (210,19819), (200,17807), (184,16703), (371,15527), (487,9967), (333,9743).
Currently, there are no known (probable) primes of the form (n^p-1)/(n-1) with odd prime p for n = 185, 269, 281, 311, 326, 380, 384, 385, 394, 396, 452, 465, 485, 511, 522, 570, 574, 598, 601, 629, 631, 632, 636, 640, 649, 670, 684, 691, 693, 711, 713, 731, 752, 759, 771, 795, 820, 861, 866, 872, 881, 932, 938, 948, 951, 956, 963, 996, 1005, 1015. (less than 1024) All of these n have been searched to at least p = 10000 with no primes found.

To be completed...