# The number of integers less than x that have at least two distinct prime factors of bit size greater than one-third the bit size of x

Sander came out with a paper describing how to generate what he calls an RSA-UFO. Anoncoin then utilizes this and mentions that the paper proves that the probability that a randomly generated integer, x, that is divisible by at least two primes of bit size greater than or equal to one-third the bit size of x is about 0.16.

However, I see in the paper the theorem saying

Let ξ $$\in (\frac{1}{3}, \frac{5}{12})$$. Then the number of integers $$\leq x$$ that have two distinct prime factors $$\geq x^ξ$$ is $$x(\frac{1}{2} \ln^2(\frac{1}{2ξ}) + O(\frac{1}{ln(x)}))$$

Taking $$O(\frac{1}{ln(x)})$$ as about 0 (since it approaches 0 as x grows), I see that the probability should be

$$\frac{1}{2} \ln^2(\frac{1}{2ξ}) \approx 0.082$$

when $$ξ = \frac{1}{3}$$, which should give a prime bit size of one-third the bit size of x. I do notice that I am off by a factor of about two, but it seems to me that that $$\frac{1}{2}$$ is necessary.

Where does the 0.16 probability come from, and where am I going wrong?

EDIT: I am wondering if I am not including the fact that generating random numbers with a particular bit size ensures the first bit is 1, which reduces the possible generated numbers less than x by half, about doubling the probability?