Proof of the reciprocal of all semiprimes diverging?

Question

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$

I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank you.

Answer 1

a subseries is $$ \frac{1}{2} \left( \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \right) $$

Answer 2

If you look just at the terms with even denominators you have half the sum of the reciprocals of the primes.

Curiously enough, for each prime $p$ the terms with denominator a multiple of $p$ give $1/p$ times the (infinite) sum of the reciprocals of the primes. Of course these subseries overlap. Euler could probably make something of this.