For integers $ngeq 1$ in this post we denote the Ramanujan primes as $R_n$, see for example the Wikipedia *Ramanujan prime* or [1]. I don’t know if my question is in the literature but I think that it should be very difficult to solve. Compare with the sequence and related literature added for the sequence *A045345* from The Encyclopedia of Integer Sequences.

Question (Edited).I would like to know if there exist infinitely many integers $ngeq 1$ such that $$nmidsum_{k=1}^n R_k.$$

Alternatively, if the question is difficult to answer, can you provide heuristic reasonings tell me about whether there are infinitely many or well a finite number of terms for this integer sequence? You can to invoke conjectures from the literature in your discussion.Many thanks.

Since I think that it is very difficult I’m asking if one can do some work about the question or add some reasoning or heuristic.

The sequence of integers $ngeq 1$ for which $frac{sum_{k=1}^n R_k}{n}$ is integer starts as $$1,3,5,63,3669,8933,ldots$$

For example (the sequence in OEIS for Ramanujan primes is *A104272*) $$frac{1}{5}left(2+11+17+29+41right)=frac{100}{5}inmathbb{Z}.$$

## References:

[1] Jonathan Sondow, John W. Nicholson and Tony D. Noe, *Ramanujan Primes: Bounds, Runs, Twins, and Gaps*, Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.2.