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A108186
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New approximation of PrimePi based on x/(log[x]-1) and Integrate[x/Log[x],{x,2,n}] starting at n=10.
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0
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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, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24
(list; graph; refs; listen; history; internal format)
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OFFSET
| 10,1
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COMMENTS
| 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}]
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FORMULA
| 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]]]
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MATHEMATICA
| 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}]
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CROSSREFS
| Sequence in context: A021853 A092616 A096251 * A024818 A076236 A072520
Adjacent sequences: A108183 A108184 A108185 * A108187 A108188 A108189
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 14 2005
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