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%I #12 Nov 13 2022 14:44:14
%S 2,4,6,9,14,19,26,33,44,55,66,77,90,103,120,137,154,171,190,209,232,
%T 255,278,301,330,359,388,417,446,475,506,537,574,611,648,685,722,759,
%U 800,841,882,923,966,1009,1056,1103,1150,1197,1250,1303,1356
%N Partial sum of smallest prime >= n (A007918).
%C 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.
%D Schechter, B., My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: Simon and Schuster, 1998.
%H Robert Israel, <a href="/A117842/b117842.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein et al., <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate.</a>
%F 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).
%e 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.
%p ListTools:-PartialSums(map(nextprime,[$-1..100])); # _Robert Israel_, Aug 09 2020
%t Accumulate[NextPrime[Range[0,50]-1]] (* _Harvey P. Dale_, Nov 13 2022 *)
%Y Cf. A000040, A007504, A007918.
%K easy,nonn
%O 0,1
%A _Jonathan Vos Post_, Apr 30 2006
%E Corrected by _T. D. Noe_, Nov 01 2006
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