## Some Flat Equation Prime Number Aproximations

for estimating Pn

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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 we are more interested to know how you derived those equations.
 Well I was investigating counting with A002260 and A004736 and tried using The triangle number formula to balance both pattern one and pattern two, the formula to which is on my website, Then I Played with the counting function with some other functions like log and then simplified with the help of the computer my best effort at predicting Pn and tweaked around with that until well what you see so far as my effort. I still have my work cut out for me to see how accurate I can get and am working towards a few tweaks with the addition of some extra functionality including making the smooth curve reverberate. All still as a flat equation. The pi(x) approximation is my first attempt (roughly 2 hours work) at that problem so it should be quite easy for me to Improve it's accuracy and give it the Pn weaving ability that the far more worked on Pn estimates have.

## Some Flat Equation Prime Number Aproximations

My Pn estimates seem weave round Pn with an upper bound of 0.075 n of Pn and a lower bound of 0.5 n of Pn all at an ever slower speed exponentially relative to Pn, which Means with a slight modification to the formula I can get with in roughly 0.28 n of Pn for any number no matter how large providing this conjecture works to infinity.
 I would suggest you post some numerical results and/or graphs to let people get a feel for how good your approximations are.
 on it way I should have that for you soon.