Is there a constant representing the Engel Expansion of the Primes?

Question

Given the set of all prime numbers $P$, is there a constant that has an Engel Expansion of $P$?

I coded a script to calculate the number, and it is about $0.705230171791801...$

Mathematically, the number can be found like this, with $k$ used as a placeholder for the number:

$$k=\lim_{n\to\infty}\sum_{i=1}^n{1\over p_i\#}={1\over2}+{1\over6}+{1\over30}...$$

Where $p_k\#$ is the primorial of the $k$th prime.

I google searched the result I got, but I found no results.

I guess I'm just wondering if anybody has come across this number before.

Answer 1

Your script is wrong (an off-by-one error). My Maple computation gives:

$$\lim_{n\to\infty}\sum_{i=1}^n{1\over p_i\#} \approx 0.70523017179180096514743168$$ If you search with inverse primorial seriesyou find among others this statement

The sum of the reciprocals is given by: $$S=1/2+1/6+1/30+1/210+1/2310+1/30030+1/510510+1/9699690\\+1/223092870+1/6469 693230+1/200560490130+1/742073813 + \dots\\ = 0,7052301717918$$ This value is very near to the spin of the final black hole produced by the collision of two blacks holes and calculated from the observations of gravitational waves.

in https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2017a.pdf.

I honestly cannot judge the truth of the last sentence.