Smallest Prime Factor

Let $smpf (n)$ be the smallest prime factor of $n$ .
$smpf (91) = 7$ because $91 = 7 \times 13$ and $smpf (45) = 3$ because $45 = 3 \times 3 \times 5$ .
Let $S (n)$ be the sum of $smpf (i)$ for $2 \leq i \leq n$ .
E.g. $S (100) = 1257$ .

Find $S (10^{12}) mod 10^{9}$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.