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A112270 One third of the sum of the first n primes, when an integer. 2
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 (list; graph; refs; listen; history; text; internal format)
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.

Moser, L. "Notes on Number Theory III. On the Sum of Consecutive Primes." Can. Math. Bull. 6, 159-161, 1963.

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

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 * A124826 A136069 A140028

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

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Last modified August 3 08:10 EDT 2020. Contains 336197 sequences. (Running on oeis4.)