Numbers not Sum of Square and Prime

Sequence
The sequence of (strictly) positive integers which is not the sum of a square and a prime begins:
 * $1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, \ldots$

Proof
$1$ is trivial.

For each further $n$ on the list, we subtract successive squares less than $n$ and show that no prime can result.

As follows: