A336235 Numbers m such that Sum_{i=3..m} (prime(i) modulo 6) = 3*m, where prime(i) is the i-th prime.

17, 33, 35, 41, 43, 45, 55, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 95, 101, 115, 117, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 181, 183, 189, 191, 193, 275, 277, 281, 283, 291, 341, 355, 521, 523, 525, 527

Offset: 1

Author

Ctibor O. Zizka, Jul 13 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Crossrefs

Cf. A000040, A039704.

Programs

Mathematica

s = Accumulate[Mod[Select[Range[5, 200000], PrimeQ], 6]]; 2 + Position[s - 3 * Range[Length[s]], 6] // Flatten (* Amiram Eldar, Jul 13 2020 *)

PARI

isok(m) = sum(i=3, m, prime(i)%6) == 3*m; \\ Michel Marcus, Jul 13 2020

Extensions

More terms from Michel Marcus, Jul 13 2020