This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365591 #40 Sep 17 2023 06:11:54
%S A365591 1,5,8,17,28,33,40,41,49,52,64,65,69,77,92,93,108,109,120,121,136,137,
%T A365591 140,144,165,200,201,204,225,229,265,269,272,280,292,312,325,332,337,
%U A365591 344,356,361,369,376,388,457,464,473,480,529,541,548,553,556,573,577
%N A365591 Numbers k such that Sum_{i=1..k} prime(i) + i is prime.
%C A365591 Numbers k such that A000217(k) + A007504(k) is prime. - _Robert Israel_, Sep 10 2023
%e A365591 2+1 = 3, which is prime, so 1 is a term.
%e A365591 2+1 + 3+2 + 5+3 + 7+4 + 11+5 = 43, which is prime, so 5 is a term.
%p A365591 P:= [seq(ithprime(i),i=1..1000)]:
%p A365591 S:= ListTools:-PartialSums(P):
%p A365591 select(i -> isprime(S[i]+i*(i+1)/2),[$1..1000]); # _Robert Israel_, Sep 10 2023
%t A365591 With[{m = 600}, Position[Accumulate[Range[m] + Prime[Range[m]]], _?PrimeQ] // Flatten] (* _Amiram Eldar_, Sep 10 2023 *)
%o A365591 (PARI) isok(k) = isprime(sum(i=1, k, i+prime(i))); \\ _Michel Marcus_, Sep 14 2023
%Y A365591 Cf. A000217, A007504, A014688.
%K A365591 nonn,easy
%O A365591 1,2
%A A365591 _Saish S. Kambali_, Sep 10 2023