--description--

A positive integer matrix is a matrix whose elements are all positive integers.

Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:

$$\begin{pmatrix} 40 & 12 \\ 48 & 40 \end{pmatrix} = {\begin{pmatrix} 2 & 3 \\ 12 & 2 \end{pmatrix}}^2 = {\begin{pmatrix} 6 & 1 \\ 4 & 6 \end{pmatrix}}^2$$

We define $F(N)$ as the number of the 2x2 positive integer matrices which have a trace less than N and which can be expressed as a square of a positive integer matrix in two different ways.

We can verify that $F(50) = 7$ and $F(1000) = 1019$.

Find $F({10}^7)$.

--hints--

positiveIntegerMatrix() should return 145159332.

assert.strictEqual(positiveIntegerMatrix(), 145159332);

--seed--

--seed-contents--

function positiveIntegerMatrix() {

  return true;
}

positiveIntegerMatrix();

--solutions--

// solution required