Splitting into Primes

A mass split is an operation on a multiset $K$ of positive integers. One mass split handles every element of $K$ at the same time. An element $k$ that is prime is left alone. An element $k$ that is not prime is replaced by the two numbers $d$ and $k/d$, where $d$ is a divisor of $k$ with $1 < d < k$. Every divisor that meets the condition is equally likely, and each element draws its divisor independently.

Take $K = \{2, 10, 12, 12\}$ as an example. 2 is prime and stays. The only divisors of 10 strictly between 1 and 10 are 2 and 5, so 10 always becomes $\{2, 5\}$. Each 12 picks one of 2, 3, 4, 6 with probability $1/4$, which turns it into $\{2, 6\}$ or $\{3, 4\}$ with probability $1/2$ each. The first mass split therefore gives $\{2, 2, 3, 3, 4, 4, 5\}$ with probability $0.25$, $\{2, 2, 2, 3, 4, 5, 6\}$ with probability $0.5$, and $\{2, 2, 2, 2, 5, 6, 6\}$ with probability $0.25$. A second mass split applied to the last of those gives $\{2, 2, 2, 2, 2, 2, 3, 3, 5\}$ and nothing else.

Start from the multiset $\{N\}$, which holds the single element $N$, and repeat the mass split until every element is prime. Compute the expected number of mass splits.

The first line contains the number of test cases $T$ ($1 \le T \le 10^4$).

Each of the next $T$ lines contains one starting value $N$ ($2 \le N \le 10^{10}$).

For each test case, print the expected number of mass splits on its own line. Round the value and print exactly six digits after the decimal point. An answer of 2 is printed as 2.000000.