Adapting the solution in page 81 of [1] of Problem 1.69 (Amer. Math. Monthly) for the sequence of Ramanujan primes $\{R_k\}_{k\geq 1}$ one deduces the following claim. Wikipedia has an article for Ramanujan primes.
Claim. For any integer $m\geq 1$ there exists an integer $N(m)$ such that $$\prod_{n=m}^{N(m)}(R_n+1)>2\prod_{n=m}^{N(m)}R_n\tag{1}$$ holds.
Sketch of proof. Just some companion details are that I've invoked the Theorem B-1 from Appendix B_1 of George Andrews, Number Theory, Dover Publications (1971), together the nature of the series $\sum_{k=1}^\infty 1/R_k$ that is a consequence of a theorem due to Sondow (see [2]).$\square$
Question. I wondered if one can to deduce some application (to get a more elaborated statement) from previous claim, that is from the inequality $(1)$. Can you get any consequence (with a good mathematical content) from previous inequality? Many thanks.
I tried to think some consequence for aliquot sequences, but without results.
References:
[1] Valentin Boju and Louis Funar, The Math Problems Notebook, Birkhäuser (2007).
[2] Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (7), pp. 630–635 (2009).