Total Inversion Count of Divided Sequences

 Published on Sunday, 8th March 2020, 04:00 am; Solved by 299;
Difficulty rating: 25%

Problem 705

The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the sequence.
For example, 34214 has inversion count of 5: $34214 \to 32414 \to 23414 \to 23144 \to 21344 \to12344$.

If each digit of a sequence is replaced by one of its divisors a divided sequence is obtained.
For example, the sequence 332 has 8 divided sequences: $\{332,331,312,311,132,131,112,111\}$.

Define $G(N)$ to be the concatenation of all primes less than $N$, ignoring any zero digit.
For example, $G(20) = 235711131719$.

Define $F(N)$ to be the sum of the inversion count for all possible divided sequences from the master sequence $G(N)$.
You are given $F(20) = 3312$ and $F(50) = 338079744$.

Find $F(10^8)$. Give your answer modulo $1\,000\,000\,007$.