A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).

0, 2, 5, 8, 11, 14, 18, 22, 26, 30, 35, 40, 45, 50, 56, 61, 67, 74, 80, 87, 94, 101, 108, 116, 123, 131, 140, 148, 157, 165, 175, 184, 193, 203, 213, 223, 233, 243, 254, 265, 276, 287, 299, 310, 322, 334, 346, 359, 371, 384, 397, 410, 424, 437, 451, 465, 479, 493

Offset: 1

Author

Cino Hilliard, Jul 30 2009

Keywords

• nonn

Comments

a(n) closely approximates the number of primes < n^2, that is, A038107(n) = Pi(n^2).

In fact, the sum is as good as Li(n^2). For n = 10^9,

a(n) = 24739954333817884.

Pi(n^2) = 24739954287740860 = A006880(18).

Li(n^2) = 24739954309690415 = A057754(18) = A089896(18).

R(n^2) = 24739954284239494 = A057793(18).

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 far from Pi(n^2), see A058290.

Examples

			For n = 10, floor(Sum_{x=2..n} x/log(x)) = 30, the 10th term.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Programs

Mathematica

Table[Floor[Sum[j/Log[j], {j, 2, n}]], {n,1,50}] (* G. C. Greubel, Jul 27 2017 *)
Join[{0},Floor[Accumulate[Table[x/Log[x],{x,2,60}]]]] (* Harvey P. Dale, May 22 2021 *)

PARI

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

Formula

a(10^n) = A163521(n).

Extensions

Offset corrected, definition detailed, 7 references to other sequences added by R. J. Mathar, Aug 29 2009