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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 far from Pi(n^2).
LINKS
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
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 February 28 21:38 EST 2024. Contains 370400 sequences. (Running on oeis4.)