## One-child Numbers

### Problem 413

Published on Sunday, 3rd February 2013, 04:00 am; Solved by 224; Difficulty rating: 75%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}).