Largest prime

Consider the sequence $n^2+3$ with $n\ge 1$. If we write down the first terms of this sequence we get: $4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, $ $147,172, 199, 228, 259, 292, 327, 364$... .

We see that the terms for $n=6$ and $n=7$($39$ and $52$) are both divisible by $13$. In fact $13$ is the largest prime dividing any two successive terms of this sequence.

Let $P(k)$ be the largest prime that divides any two successive terms of the sequence $n^2+k^2$.

Find the last $18$ digits of $\sum_{k=1}^{10\ 000\ 000}P(k)$. .