A336235 - 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