A112040 -id:A112040 - cp's OEIS frontend

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

Jonathan Vos Post, Nov 30 2005

			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.

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.

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.

Cf. A000040, A007504, A112040.

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 *)

{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.

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

A112271 One fifth of the sum of the first n primes, when an integer.

Original entry on oeis.org

1, 2, 20, 32, 88, 212, 296, 344, 1070, 1166, 1374, 1655, 2248, 2698, 3368, 3730, 3916, 4936, 5160, 5388, 6725, 6983, 8788, 11338, 12382, 12923, 13480, 15026, 16244, 17717, 19033, 19481, 19937, 21108, 24584, 29191, 30345, 33008, 33921, 34850

Offset: 1

Jonathan Vos Post, Nov 30 2005

			a(1) = 1 = (2+3)/5 = A007504(2)/5 = 5/5.
a(2) = 2 = (2+3+5)/5 = A007504(3)/5 = 10/5.
a(3) = 20 = (2+3+5+7+11+13+17+19+23)/5 = A007504(9)/5 = 100/5.
a(4) = 32 = (2+3+5+7+11+13+17+19+23+29+31)/5 = A007504(11)/5 = 160/5.
a(5) = 88 = A007504(17)/5 = 440/5.
a(6) = 212 = A007504(25)/5 = 1060/5.
a(7) = 296 = A007504(29)/5 = 1480/5.
a(8) = 344 = A007504(31)/5 = 1720/5.

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.

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.

Cf. A000040, A007504, A111319, A112040.

s = 0; lst = {}; Do[s = s + Prime[n]; If[Mod[s, 5] == 0, AppendTo[lst, s/5]], {n, 250}]; lst (* Robert G. Wilson v, Dec 04 2005 *)
Select[Accumulate[Prime[Range[400]]]/5,IntegerQ] (* Harvey P. Dale, May 03 2017 *)

{a(n)} = {A007504(k)/5 iff 5 | A007504(k)}. {a(n)} = {(p_1 + p_2 + ... + p_k)/5 iff the sum is an integer}. It is sufficient that A007504(k) == 0 (mod 10), but not necessary (the last five consecutive primes ending in 1 can give a solution). It is necessary that k = 2 or k is odd.

More terms from Stefan Steinerberger and Robert G. Wilson v, Dec 04 2005

A112272 One seventh of the sum of the first n primes, when an integer.

Original entry on oeis.org

4, 11, 34, 113, 284, 441, 634, 731, 943, 1226, 1657, 2343, 2469, 3767, 3931, 4271, 4712, 5759, 5963, 7154, 8221, 8467, 8971, 9362, 9763, 12655, 13279, 13595, 13915, 15941, 17560, 19641, 21261, 21675, 22091, 22937, 23363, 23793, 24671, 26702

Offset: 1

Jonathan Vos Post, Nov 30 2005

a(1) = 4 and a(6) = 441 are perfect squares. What is the next term in this subsequence? Answer from Stefan Steinerberger: a(103)=315844=562^2.

			a(1) = 4 = (2+3+5+7+11)/7 = A007504(5)/7 = 28/7.
a(2) = 11 = (2+3+5+7+11+13+17+19)/5 = A007504(8)/7 = 77/7.
a(3) = 34 = A007504(13)/5 = 238/7.
a(4) = 113 = A007504(22)/5 = 791/7.
a(5) = 284 = A007504(33)/5 = 1988/7.
a(6) = 441 = A007504(40)/5 = 3087/7.

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.

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.

Cf. A000040, A007504, A112040.

s = 0; lst = {}; Do[s = s + Prime[n]; If[Mod[s, 7] == 0, AppendTo[lst, s/7]], {n, 270}]; lst (* Robert G. Wilson v *)
Select[Accumulate[Prime[Range[300]]]/7,IntegerQ] (* Harvey P. Dale, Nov 26 2014 *)

{a(n)} = {A007504(k)/7 iff 7 | A007504(k)}. {a(n)} = {(p_1 + p_2 + ... + p_k)/7 iff the sum is an integer}.

More terms from Stefan Steinerberger and Robert G. Wilson v, Dec 02 2005

cp's OEIS Frontend

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

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A112271 One fifth of the sum of the first n primes, when an integer.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A112272 One seventh of the sum of the first n primes, when an integer.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions