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%I #3 Mar 30 2012 17:34:19
%S 7,7,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,12,13,13,13,
%T 13,14,14,14,14,15,15,15,15,16,16,16,16,16,17,17,17,17,18,18,18,18,18,
%U 19,19,19,19,20,20,20,20,20,21,21,21,21,21,22,22,22,22,23,23,23,23,23,24
%N New approximation of PrimePi based on x/(log[x]-1) and Integrate[x/Log[x],{x,2,n}] starting at n=10.
%C A singularity exists at low n value. Average error in 10 to 250 is 1.51867 error = Table[Floor[Sqrt[(y /. NSolve[( x/(Log[x] - 1) + y)/100 - Sqrt[y*x/( Log[x] - 1)]/10 == 0, y][[1]])*(y /. NSolve[(x/(Log[ x] - 1) + y)/100 - Sqrt[y*x/(Log[x] - 1)]/10 == 0, y][[2]])]] - PrimePi[x], {x, 10, 250}]
%F x1=x/(Log[x]-1) y1=Integrate[x/Log[x], {n, 2, n}] f[n]=Solve[(x1+y1)/100==(x1*y1)^(1/2)/10, y1] a(n) = Sqrt[f[n][[1]]*f[[n][[2]]]
%t PiN = Table[Floor[Sqrt[(y /. NSolve[(x/(Log[x] - 1) + y)/100 - Sqrt[y*x/(Log[x] - 1)]/10 == 0, y][[1]])*(y /. NSolve[(x/(Log[x] - 1) + y)/100 - Sqrt[y*x/(Log[x] - 1)]/10 ==0, y][[2]])]], {x, 10, 250}]
%K nonn,uned
%O 10,1
%A _Roger L. Bagula_, Jun 14 2005
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