Eccoci.

We say that a $d$-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divisible by $d$.

For example, $5671$ is a $4$-digit one-child number. Among all its sub-strings $5$, $6$, $7$, $1$, $56$, $67$, $71$, $567$, $671$ and $5671$, only $56$ is divisible by $4$.
Similarly, $104$ is a $3$-digit one-child number because only $0$ is divisible by $3$.
$1132451$ is a $7$-digit one-child number because only $245$ is divisible by $7$.

Let $F(N)$ be the number of the one-child numbers less than $N$.
We can verify that $F(10) = 9$, $F(10^3) = 389$ and $F(10^7) = 277674$.

Find $F(10^{19})$.