Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #19 Jul 14 2020 22:53:00
%S 17,33,35,41,43,45,55,59,61,63,65,67,69,71,73,77,79,81,83,87,89,91,93,
%T 95,101,115,117,131,133,135,137,139,141,143,145,147,149,151,153,155,
%U 157,159,181,183,189,191,193,275,277,281,283,291,341,355,521,523,525,527
%N Numbers m such that Sum_{i=3..m} (prime(i) modulo 6) = 3*m, where prime(i) is the i-th prime.
%C By the Prime Number Theorem for arithmetic progressions, all nonzero residue classes are equiprobable. In particular, asymptotically, as m -> oo the Sum_{i=r..m} (prime(i) modulo k) = m*k/2. For this sequence this says Sum_{i=3..m} (prime(i) modulo 6) = m*3.
%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%e For m = 17 we have Sum_{i=3..17} (prime(i) modulo 6) = 5 + 1 + 5 + 1 + 5 + 1 + 5 + 5 + 1 + 1 + 5 + 1 + 5 + 5 + 5 = 3*17.
%t s = Accumulate[Mod[Select[Range[5, 200000], PrimeQ], 6]]; 2 + Position[s - 3 * Range[Length[s]], 6] // Flatten (* _Amiram Eldar_, Jul 13 2020 *)
%o (PARI) isok(m) = sum(i=3, m, prime(i)%6) == 3*m; \\ _Michel Marcus_, Jul 13 2020
%Y Cf. A000040, A039704.
%K nonn
%O 1,1
%A _Ctibor O. Zizka_, Jul 13 2020
%E More terms from _Michel Marcus_, Jul 13 2020