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 A140208 Floor n*Pi(n)/2. 0
 0, 1, 3, 4, 7, 9, 14, 16, 18, 20, 27, 30, 39, 42, 45, 48, 59, 63, 76, 80, 84, 88, 103, 108, 112, 117, 121, 126, 145, 150, 170, 176, 181, 187, 192, 198, 222, 228, 234, 240, 266, 273, 301, 308, 315, 322, 352, 360, 367, 375, 382, 390, 424, 432, 440, 448, 456, 464 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS I introduce this sequence which is A128913(n)/2 because it is closely related to the prime counting function Pi(n) and the sum of primes < n for large n. This is, SumP(n) ~ n*Pi(n)/2. For n = 10^10 n*Pi(n)/2 = 2275262555000000000. Sum primes < 10^n = 2220822432581729238. This has error 0.0245...For the largest known sum of primes, for sums < 10^20, we have n*Pi(n)/2 = 111040980128045942000000000000000000000. The sum of primes < 10^20 = 109778913483063648128485839045703833541. The error here is -0.01149... It converges quite slowly and better approximations have been found. This relationship was derived by using the summation formula for an arithmetic progression. For the odd integers where n is even, let the first term = 1, the last term is n-1 and the number of terms is n/2. So the sum of the odd numbers < n is ((1 +n-1)*n/2)/2. If we let Pi(x) be the number of terms, we get the result n*Pi(n)/2. A closed formula, SumP(n) ~ n^2/(2*log(n)-1) is quite accurate. The best formula I have found is the remarkable SumP(n) ~ Pi(n^2). This formula has an error of 6.162071097138 E-11 for the largest known sum of primes or sum < 10^20. Proof: 2+3+..+prime(n) = A007504(n) ~ n^2 log n / 2 (Bach and Shallit, 1996). Let n = Pi(x) ~ x/log x. So A007504(n) ~ (x/log x)^2 log(x/log x) / 2 ~ x^2 / (2 log x) ~ Pi(x^2). QED. - Thomas Ordowski, Aug 12 2012. See the link Sum of Primes for derivations of these asymptotic formulas. LINKS Cino Hilliard, Sum of Primes. FORMULA Pi(n) is the prime counting function, the number of primes < n. Define SumP(n) is the sum of primes < n. PROG (PARI) g(n) = for(x=1, n, print1(floor(x*primepi(x)/2)", ")) CROSSREFS Cf. A128913. Sequence in context: A146994 A295069 A103054 * A098390 A266769 A008763 Adjacent sequences:  A140205 A140206 A140207 * A140209 A140210 A140211 KEYWORD nonn,uned AUTHOR Cino Hilliard, Jun 09 2008 STATUS approved

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Last modified November 17 05:27 EST 2019. Contains 329217 sequences. (Running on oeis4.)