A112270 One third of the sum of the first n primes, when an integer.

43, 127, 167, 213, 321, 387, 457, 531, 617, 709, 809, 1029, 1149, 1277, 1409, 1863, 2027, 2290, 3397, 3629, 4113, 4367, 4629, 4899, 5179, 5467, 5761, 6063, 6371, 7516, 7864, 8600, 8980, 9368, 10168, 10578, 11856, 12296, 12746, 13204, 13674, 14156

Offset

1,1

References

Bach, E. and Shallit, J. Sect. 2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, 1996.

H. L. Nelson, "Prime Sums", J. Rec. Math., 14 (1981), 205-206.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161.

Eric Weisstein's World of Mathematics, Prime Sums.

Formula

{a(n)} = {A007504(k)/3 iff 3 | A007504(k)}. {a(n)} = {(p_1 + p_2 + ... + p_k)/3 iff the sum is an integer}. It is necessary but not sufficient for k to be even.

Example

a(1) = 43 = (2+3+5+7+11+13+17+19+23+29)/3 = A007504(10)/3 = 129/3.
a(2) = 127 = A007504(16)/3 = 381/3.
a(3) = 167 = A007504(18)/3 = 501/3.
a(4) = 213 = A007504(20)/3 = 639/3.
a(5) = 321 = A007504(24)/3 = 963/3.
a(6) = 387 = A007504(26)/3 = 1161/3.

Mathematica

s = 0; lst = {}; Do[s = s + Prime[n]; If[Mod[s, 3] == 0, AppendTo[lst, s/3]], {n, 130}]; lst (* Robert G. Wilson v *)
Select[Accumulate[Prime[Range[200]]]/3, IntegerQ] (* Harvey P. Dale, Feb 20 2018 *)

Crossrefs

Cf. A000040, A007504, A112040.

Sequence in context: A029816 A044294 A044675 * A356510 A124826 A136069

Adjacent sequences: A112267 A112268 A112269 * A112271 A112272 A112273

Keyword

easy,nonn

Author

Jonathan Vos Post, Nov 30 2005

Extensions

More terms from Robert G. Wilson v, Nov 30 2005

Status

approved