Primes on "corners" are connected with a line. A corner, in the case of a Ulam Spiral, is like corners in real life. It's where we turn to go in a different direction. The corner primes in this image are:

2,3,5,7,13,17,31,37,43,73,101,157,197,211,241,257,307,401,421,463,577,601,677,757,1123,1297, and 1483.

Now if you take the average of these numbers you get 242.074074074.... Subtract 1 because I said so, and you get 241.074074074..., which rounds to 241. Prime!

Do the same with 2,3,5,7,13,17. Averages to 7.8 - 1 = 6.8. Rounds to 7. Bingo. Prime!

Do it with 2,3,5,7,13,17, and 31. Averages to 11.142857. Don't subtract the 1 because then our rounded average would not be prime. Dang it.

Weird? Yes. Significant? Nope.

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