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The Wolfram MathWorld page for "Semiprime" (k=2) at http://mathworld.wolfram.com/Semiprime.html gives the following formula:

"A formula for the number of semiprimes less than or equal to $n$ is given by

$\pi_2(x) = \sum_{k=1}^{\pi(\sqrt{x})} [\pi(x/p_k)-k+1]$

where $\pi(x)$ is the prime counting function and $p_k$ is the $k$th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006)."

Curiously, the number of terms in the above sum, $\pi(\sqrt{x})$, is approximately $Li(\sqrt{x})$, which is the order of the main error term in the formula $\pi(x) = Li(x) + e(x)$ itself. Further, the value of $\pi(x/p_k)$ in the final term is also equal to $\pi(x/\sqrt{x}) = \pi(\sqrt{x})$.

Here is a follow-up question: Has there been any research about the distribution of primes and semiprimes together? One would expect the error term to be smaller than for either primes or semiprimes alone, because a region with fewer primes will tend to have more semiprimes, and vice versa.

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The Wolfram MathWorld page for "Semiprime" (k=2) at http://mathworld.wolfram.com/Semiprime.html gives the following formula:

"A formula for the number of semiprimes less than or equal to is given by

$\pi_2(x) = \sum_{k=1}^{\pi(\sqrt{x})} [\pi(x/p_k)-k+1]$

where $\pi(x)$ is the prime counting function and $p_k$ is the $k$th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006)."

Curiously, the number of terms in the above sum, $\pi(\sqrt{x})$, is approximately $Li(\sqrt{x})$, which is the order of the main error term in the formula $\pi(x) = Li(x) + e(x)$ itself. Further, the value of $\pi(x/p_k)$ in the final term is also equal to $\pi(x/\sqrt{x}) = \pi(\sqrt{x})$.

Here is a follow-up question: Has there been any research about the distribution of primes and semiprimes together? One would expect the error term to be smaller than for either primes or semiprimes alone, because a region with fewer primes will tend to have more semiprimes, and vice versa.