

A108186


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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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
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OFFSET

10,1


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}]


LINKS

Table of n, a(n) for n=10..82.


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]]]


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}]


CROSSREFS

Sequence in context: A260634 A294968 A096251 * A024818 A076236 A072520
Adjacent sequences: A108183 A108184 A108185 * A108187 A108188 A108189


KEYWORD

nonn,uned


AUTHOR

Roger L. Bagula, Jun 14 2005


STATUS

approved



