1,1

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 fo large n. For n = 10^9,

a(n) = 24739954333817884.

Pi(n^2) = 24739954287740860.

Li(n^2) = 24739954309690415. Logarithmic integral approximation of Pi(n^2)

R(n^2) = 24739954284239494. Riemann's approximation of Pi(n^2)

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

10^18/(log(10^18)-1)=24723998785919976 and 10^18/log(10^18)=24127471216847323.

Ironically though, a(n) = sum(x=2,n,x/(log(x)-1) is way off Pi(n^2).

For n = 9, floor(sum(x=2,10^n,x/log(x))) = 24739954333817884, the 9-th entry.

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

Sequence in context: A027488 A075187 A060076 this_sequence A002456 A107768 A048536

Adjacent sequences: A163518 A163519 A163520 this_sequence A163522 A163523 A163524

nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Jul 30 2009

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