A117842 Partial sum of smallest prime >= n (A007918).

2, 4, 6, 9, 14, 19, 26, 33, 44, 55, 66, 77, 90, 103, 120, 137, 154, 171, 190, 209, 232, 255, 278, 301, 330, 359, 388, 417, 446, 475, 506, 537, 574, 611, 648, 685, 722, 759, 800, 841, 882, 923, 966, 1009, 1056, 1103, 1150, 1197, 1250, 1303, 1356

Offset

0,1

Comments

Bertrand's [1845] postulate as proved by Chebyshev [1850] is versified: "Chebyshev said it, but I'll say it again; There's always a prime between n and 2n." [N. J. Fine in Schechter, 1998]. This sequence is the partial sum of the least such primes. It differs from A007504 "sum of first n primes" because of the repetitions in A007918.

References

Schechter, B., My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: Simon and Schuster, 1998.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Eric Weisstein et al., Bertrand's Postulate.

Formula

a(n) = SUM[i=0..n] A007918(n). a(n) = SUM[i=0..n] smallest prime >= i. a(n) = SUM[i=0..n] nextprime(i).

Example

a(50) = 2+ 2+ 2+ 3+ 5+ 5+ 7+ 7+ 11+ 11+ 11+ 11+ 13+ 13+ 17+ 17+ 17+ 17+ 19+ 19+ 23+ 23+ 23+ 23+ 29+ 29+ 29+ 29+ 29+ 29+ 31+ 31+ 37+ 37+ 37+ 37+ 37+ 37+ 41+ 41+ 41+ 41+ 43+ 43+ 47+ 47+ 47+ 47+ 53+ 53+ 53 = 1356.

Maple

ListTools:-PartialSums(map(nextprime, [$-1..100])); # Robert Israel, Aug 09 2020

Mathematica

Accumulate[NextPrime[Range[0, 50]-1]] (* Harvey P. Dale, Nov 13 2022 *)

Crossrefs

Cf. A000040, A007504, A007918.

Sequence in context: A070211 A113753 A024457 * A067588 A003402 A328863

Adjacent sequences: A117839 A117840 A117841 * A117843 A117844 A117845

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 30 2006

Extensions

Corrected by T. D. Noe, Nov 01 2006

Status

approved