A140208 a(n) = floor(n*Pi(n)/2).

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

Offset: 1

Author

Cino Hilliard, Jun 09 2008

Keywords

• nonn
• uned

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.

Crossrefs

Cf. A007504, A128913.

Programs

Mathematica

a[n_]:=Floor[n*PrimePi[n]/2];Array[a,58] (* James C. McMahon, Jul 06 2025 *)

PARI

g(n) = for(x=1,n,print1(floor(x*primepi(x)/2)","))

Formula

Pi(n) is the prime counting function, the number of primes < n. Define SumP(n) is the sum of primes < n.