Mfcttrf

I am not sure, if this is a research problem. If not I will move this question to ME:
Let $Omega(n) = sum_p v_p(n)$, which we might view as a random variable.
Let $E_n = frac1n sum_k=1^nOmega(k)$ be the expected value and $V_n=frac1n sum_k=1^n(E_n-Omega(k))^2$ be the variance.
Then
$$pi(n) approx fracngamma(fracV_nE_n,1.4854177cdot fracV_nE_n^2)Gamma(fracV_nE_n)$$
where $Gamma=$ Gamma function, $gamma=$ lower incomplete gamma function.

Background:
I was trying to fit the gamma distribution to the random variable $Omega(k)$ ,$1 le k le n$. The value $1.4854177$ is fitted to some data.
My question is, if there is any heuristic why this approximation should be good, if at all, or if this is just an overfitting problem?

Below you can find some sage code which implements this:

def Omega(n):
 return sum([valuation(n,p) for p in prime_divisors(n)])

means = []
variances = []
xxs = []
omegas = [Omega(k) for k in range(1,10^4)]
for nn in range(10^4,10^4+3*10^3+1):
 n = nn
 omegas.append(Omega(n))
 print "---"
 m = mean(omegas[1:-1])
 v = variance(omegas[1:-1])
 shape,scale = m^2/v,v/m
 xx = find_root(lambda xx : n*(lower_gamma(shape,xx*1/scale)/gamma(shape) ).N()-prime_pi(n),1,2)
 xx = 1.4854177706344873
 approxPrimePi2 = n*(lower_gamma(shape,xx*1/scale)/gamma(shape) ).N()
 primepi = prime_pi(n)
 print primepi, approxPrimePi2,shape.N(),scale.N(),xx
 print "---"
 print "err2 = %s" % (abs(primepi-approxPrimePi2)/primepi)
 xxs.append(xx)
 means.append(m.N())
 variances.append(v.N())

Your heuristic approximation is not correct. It was proved by Turán (1934) that
$E_n$ and $V_n$ are both asymptotically $loglog n$. As a result, the RHS of your display is
$$nfracgammaleft(1+o(1),frac1.4854177+o(1)loglog nright)Gammabigl(1+o(1)bigr)=nfrac1.4854177+o(1)loglog n.$$
On the other hand, $pi(n)$ is asymptotically $n/log n$ by the prime number theorem.