Is this old news? $\sum_{i=1}^n \frac{i!}{p_i\#} \approx 1.240053652689\dots$

Question

This is a soft question as it arises out of my curiosity alone. I noticed that as $n$ increases, $\frac{n!}{p_n\#}$ decreases in magnitude much faster than $\frac{1}{p_n}$, and I wondered if the sum $$\sum_{i=1}^n \frac{i!}{p_i\#}$$ was convergent. A preliminary calculation showed that the sum very rapidly converges to $1.240053652689\dots$

A quick search for this number and did not find any reports of it. My question is: Is this number a known constant, or can anyone suggest an algebraic relationship of this number to other know constants? Please don't waste a lot of time looking for abstruse expressions; just respond if you happen to recognize this number.

Answer 1

in terms of justifying convergence: $$i!=i(i-1)(i-2)...1$$ $$p_i\#=p_{i-1}\#...(5)(3)(2)(1)$$ you could probably find a nice explicit way of comparing the orders of these two functions using $\operatorname{li}(x)\approx\pi(x)$

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As you said, OEIS returned no results nor did inverse symbolic calculator. I would certainly be interested if there is any relationship to other functions or properties of it (transcendental etc)

Answer 2

Just for the fun !

Wtih an absolute error of $4.45\times 10^{-22}$, this number corresponds to the positive root of the pentic polynomial $$284 x^5+1154 x^4-1342 x^3+571 x^2-1502 x-18=0$$ which is not found by the $ISC$.