# Math Genius: On integers \$ngeq 1\$ for which \$n\$ divides \$sum_{k=1}^n R_k\$, where \$R_k\$ denotes the \$k\$-th Ramanujan prime

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.