A163521 -id:A163521 - cp's OEIS frontend

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

Cino Hilliard, Jul 30 2009

nonn

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.

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

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

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 *)

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

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

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

A163568 Floor of the integral of x/log(x) for x=2 to 10^n.

Original entry on oeis.org

27, 1243, 78624, 5762206, 455055611, 37607950277, 3204942065688, 279238344248553, 24739954309690412, 2220819602783663480, 201467286691248261495, 18435599767366347775141, 1699246750872593033005720, 157589269275974838158399968, 14692398897720447639079087666

Offset: 1

Cino Hilliard, Jul 30 2009

nonn

This integral is an approximation to the sum of x/log(x) for x=2 to 10^n and closely approximates Pi(n^2) for x = 2 to n.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A163521.

Table[Floor[NIntegrate[x/Log[x], {x, 2, 10^n}]], {n, 1, 10}] (* G. C. Greubel, Jul 28 2017 *)

g(n)=intnum(x=2,10^n,x/log(x)); for(x=1,20,print1(floor(g(x))","))

Edited (but not checked) by N. J. A. Sloane, Aug 01 2009

Offset and upper limit in the definition corrected by R. J. Mathar, Aug 02 2009

cp's OEIS Frontend

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

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