Non-Square Positive Integers not Sum of Square and Prime

Conjecture
The sequence of (strictly) positive integers which are not square and not the sum of a square and a prime is believed to be complete:
 * $10, 34, 58, 85, 91, 130, 214, 226, 370, 526, 706, 730, 771, 1255, 1351, 1414, 1906, 2986, 3676, 9634, 21679$

Progress
From Square of n such that 2n-1 is Composite is not Sum of Square and Prime, $n^2$ is the sum of a square and a prime $2 n - 1$ composite.

Hence the question is specifically about non-squares.

No prime number is in this sequence, as trivially:
 * $p = p + 0^2$

and so is the sum of a prime (itself), and $0^2$, which is square.

Each non-square composite $n \in \Z$ can be tested by subtracting successive squares less than $n$ and investigating whether a prime can result.

In the following, the smallest $m$ such that $n - m^2 = p$ is shown where such a $p$ exists.

Otherwise the nonexistence of such a $p$ is demonstrated explicitly.

As follows:

$10$ cannot be expressed as $10 = m^2 + p$.

Thus $10$ is seen to be in this sequence.

$34$ cannot be expressed as $34 = m^2 + p$.

Thus $34$ is seen to be in this sequence.

Similarly:

$58$ cannot be expressed as $58 = m^2 + p$.

Thus $58$ is seen to be in this sequence.

This establishes the pattern.

The algorithm for determining whether a particular $n$ belongs to this sequence can be defined in pseudocode as follows: For n := 1, loop indefinitely, incrementing by 1: Is n prime? If so, continue to the next n Is n square? If so, continue to the next n For m := 1, incrementing by 1 until m^2 > n:    Is n - n^2 prime? If so, continue to the next n Next m  Add n to the sequence Next n