A163521 a(n) = floor(Sum_{k = 2..10^n} k/log(k)).

30, 1255, 78698, 5762750, 455059956, 37607986470, 3204942375900, 279238346962895, 24739954333817884

Offset

1,1

Comments

a(n) = Sum_{x=2..n} x/log(x) closely approximates the number of primes < n^2.

In fact, the sum is as good as Li(n^2) but summing a(n) is rather time consuming for large n.

For n = 10^9,

a(n) = 24739954333817884,

Pi(n^2) = 24739954287740860,

Li(n^2) = 24739954309690415,

R(n^2) = 24739954284239494,

where Li = Logarithmic integral approximation of Pi, and R = Riemann's approximation of Pi.

Now x/(log(x)-1) is a much better approximation of Pi(x) than x/log(x):

10^18/(log(10^18)-1) = 24723998785919976,

10^18/log(10^18) = 24127471216847323.

Ironically, though, a(n) = Sum_{x=2..n} x/(log(x)-1) is far from Pi(n^2).

Links

Table of n, a(n) for n=1..9.

Example

For n = 9, floor(Sum_{x=2..10^n} x/log(x)) = 24739954333817884, the 9th term.

Mathematica

Table[Floor[Sum[j/Log[j], {j, 2, 10^n}]], {n, 1, 9}] (* G. C. Greubel, Jul 27 2017 *)

Prog

(PARI) nthsum(n) = for(j=1, n, print1(floor(sum(x=2, 10^j, x/log(x)))", "));

Crossrefs

Sequence in context: A269471 A060076 A174716 * A273416 A002456 A358163

Adjacent sequences: A163518 A163519 A163520 * A163522 A163523 A163524

Keyword

nonn

Author

Cino Hilliard, Jul 30 2009

Extensions

Definition clarified by R. J. Mathar and Omar E. Pol, Aug 01 2009

Status

approved