Stern prime

A Stern prime, named for Moritz Abraham Stern, is a prime number that is not the sum of a smaller prime and twice the square of a non zero integer. Or, to put it algebraically, if for a prime q there is no smaller prime p and nonzero integer b such that q = p + 2b², then q is a Stern prime. The known Stern primes are

So, for example, if we try subtracting from 137 the first few squares doubled in order, we get {135, 129, 119, 105, 87, 65, 39, 9}, none of which is prime. That means that 137 is a Stern prime. On the other hand, 139 is not a Stern prime, since we can express it as 137 + 2(1²), or 131 + 2(2²), etc.

In fact, many primes have more than one representation of this sort. Given a twin prime, the larger prime of the pair has, if nothing else, a Goldbach representation of p + 2(1²). And if that prime is the largest of a prime quadruplet, p + 8, then p + 2(2²) is also available. Sloane's   lists odd numbers with at least n Goldbach representations. Leonhard Euler observed that as the numbers get larger, they get more representations of the form ${\displaystyle p+2b^{2}}$, suggesting that there might be a largest number with zero such representations.

...
Wikipedia