

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
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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 n1 and the number of terms is n/2. So the sum of the odd numbers < n is ((1 +n1)*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 E11 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

Table of n, a(n) for n=1..58.
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



