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

%I A163516 #16 May 22 2021 19:55:15
%S A163516 0,2,5,8,11,14,18,22,26,30,35,40,45,50,56,61,67,74,80,87,94,101,108,
%T A163516 116,123,131,140,148,157,165,175,184,193,203,213,223,233,243,254,265,
%U A163516 276,287,299,310,322,334,346,359,371,384,397,410,424,437,451,465,479,493
%N A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).
%C A163516 a(n) closely approximates the number of primes < n^2, that is, A038107(n) = Pi(n^2).
%C A163516 In fact, the sum is as good as Li(n^2). For n = 10^9,
%C A163516   a(n)    = 24739954333817884.
%C A163516   Pi(n^2) = 24739954287740860 = A006880(18).
%C A163516   Li(n^2) = 24739954309690415 = A057754(18) = A089896(18).
%C A163516   R(n^2)  = 24739954284239494 = A057793(18).
%C A163516 Now x/(log(x)-1) is a much better approximation of Pi(x) than x/log(x).
%C A163516   10^18/(log(10^18)-1) = 24723998785919976 and
%C A163516   10^18/log(10^18)     = 24127471216847323.
%C A163516 Ironically though, a(n) = Sum_{x=2..n} x/(log(x)-1) is far from Pi(n^2), see A058290.
%H A163516 G. C. Greubel, <a href="/A163516/b163516.txt">Table of n, a(n) for n = 1..1000</a>
%F A163516 a(10^n) = A163521(n).
%e A163516 For n = 10, floor(Sum_{x=2..n} x/log(x)) = 30, the 10th term.
%t A163516 Table[Floor[Sum[j/Log[j], {j, 2, n}]], {n,1,50}] (* _G. C. Greubel_, Jul 27 2017 *)
%t A163516 Join[{0},Floor[Accumulate[Table[x/Log[x],{x,2,60}]]]] (* _Harvey P. Dale_, May 22 2021 *)
%o A163516 (PARI) nthsum(n) = for(j=1,n,print1(floor(sum(x=2,j,x/log(x)))","));
%K A163516 nonn
%O A163516 1,2
%A A163516 _Cino Hilliard_, Jul 30 2009
%E A163516 Offset corrected, definition detailed, 7 references to other sequences added by _R. J. Mathar_, Aug 29 2009