Eccoci.

The arithmetic derivative is defined by

• $p^{\prime }=1$ for any prime $p$
• $(ab{)}^{\prime }=a^{\prime }b+a{b}^{\prime }$ for all integers $a,b$ (Leibniz rule)

For example, ${20}^{\prime }=24$.

Find $\sum \mathbf{gcd}(k,k^{\prime })$ for $1<k\le 5\times {10}^{15}$.

Note: $\mathbf{gcd}(x,y)$ denotes the greatest common divisor of $x$ and $y$.