sequences and series – Why does the sum of power of primes yield $ ln(frac{pi^2}{6})$?

The source for this problem is this 3b1b video. I understand this:


Now he alters the series to include only primes and powers of primes (eg. 4 and 8 are included because they are powers of 2, which is prime) while scaling down the powers of primes by a factor of the exponent, as in:


This happens to equate to: $$ ln(frac{pi^2}{6})$$

I tried to search the proof of this for a while, but could not find anything. I apologize if this happens to be a very trivial question for this site, but would be delighted to see an elementary explanation, because I am just an interested layman in this matter.