Puzzle of the Week #162 - Hypotenuse Connect Four

Some numbers can be the hypotenuse of an integer triangle, and some cannot. For instance, 24 cannot, neither can 27, but 25 and 26 both can (for example 7,24,25 and 10,24,26). In fact 25 and 26 is the first case where two consecutive numbers can each be hypotenuses. The first case of three numbers in a row which can be hypotenuses is 39, 40 and 41 (eg 15,36,39; 24,32,40; 9,40,41).

Which numbers are the first example of four in a row that could each be hypotenuse?