for estimating Pn(adsbygoogle = window.adsbygoogle || []).push({});

1/2*(8-8.7*n-n^2+1/2*(2*abs(ln(n)/ln(3)+ln(ln(n)/ln(2))/ln(2))+abs((ln(ln(3))-ln(ln(n))+2*n*ln(ln(n)/ln(2))+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln(n)))*ln(ln(n)/ln(2))))/ln(ln(n)/ln(2))))*(-1+abs(ln(n)/ln(3)+ln(ln(n)/ln(2))/ln(2))+abs(-(1/2)+n+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln(n)))*ln(ln(n)/ln(2)))/(2*ln(ln(n)/ln(2))))))

or

1/2*(3-(8+ln(2.3))*n-n^2+1/2*(-1+abs(-(1/2)+n+sqrt(ln(ln(n)/ln(2))*(-ln(ln(2))+ln(ln(n))+(8*ln(3)*ln((n*ln(8*n))/ln(n)))/ln(2)))/(2*ln(ln((n*ln(8*n))/ln(n))/ln(2))))+abs(ln(n)/ln(3)+ln(ln((n*ln(8*n))/ln(n))/ln(2))/ln(2)))*(2*abs(ln((n*ln(8*n))/ln(n))/ln(3)+ln(ln((n*ln(8*n))/ln(n))/ln(2))/ln(2))+abs(1/ln(ln(n)/ln(2))*(ln(ln(3))-ln(ln(n))+2*n*ln(ln(n)/ln(2))+sqrt(((8*ln(3)*ln(n))/ln(2)-ln(ln(2))+ln(ln((n*ln(8*n))/ln(n))))*ln(ln((n*ln(8*n))/ln(n))/ln(2)))))))

And for approximating the Pi(n)

1/(3*abs(ln(n)))*((2-n*(n-ln(n))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi))/(1+abs(ln(n)))+1/(abs(ln(n))+ln(2))*ln(2)^2*(2-n*(n-ln(n)/ln(2))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi))+1/(abs(ln(n))+ln(3))*ln(3)^2*(2-n*(n-ln(n)/ln(3))+(-1+abs(n)+abs(ln(ln(n)))/ln(pi))*(abs(n)+abs(ln(ln(n)))/ln(pi))-(2*ln(ln(n)))/ln(pi)))

I do apologise If you want more advanced formula's with Summation's and or Factorial's and or big O notation but I don't understand that stuff very well.

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# Some Flat Equation Prime Number Aproximations

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