This incipient-sequence comment in T.D. Noe's A138290 recently caught my eye. So, prime p such that 2^(p+1)-2^q-1 is composite for all positive q < p. Carlos Rivera had it published as Puzzle 437 in 2008. Therein, Giovanni Resta notes that he had checked up to 15373 without finding a third term. All of my processors were busy of course but I hadn't been using Catherine's computer in a while (too slow for my needs), so...
This morning (after several weeks of number crunching) I noticed that her machine had come up with the third term: 29251. To give you an idea of the computing involved: on my latest iMac, p = 577 (174-digit numbers) verified compositeness for all terms immediately; p = 5569 (1677-digit numbers) took a minute; and p = 29251 (8806-digit numbers) needed more than four hours!
[The sequence became A278740 on 27 Nov 2016.]
Monday, May 9
The part of Toronto in which I live used to be a town called Weston. I live in the south part of Weston, just northwest of a community called Mount Dennis. This photograph was taken in the very northern part thereof, looking down Weston Road towards Toronto's CN Tower. My camera is at maximum zoom, giving the intersection at Jane Street (200 meters away) and the streetscape beyond a seriously foreshortened look. The Tower is 10.8 km distant. I was on my way to Jane's Walk.