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A163521 a(n) = floor(Sum_{k = 2..10^n} k/log(k)). 2
30, 1255, 78698, 5762750, 455059956, 37607986470, 3204942375900, 279238346962895, 24739954333817884 (list; graph; refs; listen; history; text; internal format)
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 way off 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 entry.

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 A107768

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

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Last modified August 18 15:50 EDT 2019. Contains 326108 sequences. (Running on oeis4.)