> 23, 1039, 403026797, ... I've just heard from Phil Carmody, whom I asked for help in extending this sequence. After noting a 3/5 "near miss" with 11574185839, he found seven 4/4 lonely primes: 121829611399, 556987575337, 2015042889389, 2512979281387, 5935507174187, 8879406856379, & 11627861832299. So, 23, 1039, 403026797, 121829611399, ... Thank you, Phil!