Harmonic variations

It is well known that the Harmonic serie $\sum _{n\ge 1}\frac{1}{n}$ is divergent.

A more surprising result is that if we remove all numbers containing at least one digit 9 the corresponding serie is convergent.

What is $\sum _{n\in S_{1,2,1,4}}\frac{1}{n}$

where $S_{1,2,1,4}$ is the set of all numbers with exactly one digit 2 and one digit 4 in their base-10 representation

$S_{1,2,1,4}=\{24,42,124,142,204,214,234,240,241,243,245\dots \}$

Answer format: Give the result with the first 20 digits after the decimal point.

[My timing : < 1 sec]