The Chtholly Tree is a data structure used for certain sequence maintenance problems. Each node in a Chtholly Tree represents a consecutive block of equal elements in the sequence. These nodes are maintained in any type of sorted data structure such as set in C++ or a balanced binary search tree.
For example, the Chtholly Tree of the sequence $$$[1,2,3,3,2,2,1,1]$$$ will contain the following nodes: $$$[1,1]$$$, $$$[2,2]$$$, $$$[3,4]$$$, $$$[5,6]$$$, $$$[7,8]$$$.
Problems solvable using Chtholly Tree usually involve the following type of operation: choose a segment $$$[l,r]$$$ of the sequence, and for all $$$l \le i \le r$$$, set $$$a_i=x$$$ for some value $$$x$$$. It can be proven that if $$$l$$$, $$$r$$$, and $$$x$$$ are chosen randomly, the expected number of nodes in a Chtholly Tree is $$$O(\log n)$$$, where $$$n$$$ is the length of the sequence.
However, Baozii would like to know the exact expected number of nodes rather than its asymptotic representation. To do so, he did the following experiment:
After a sufficiently large number of operations, Baozii observed that the number of nodes seems to converge to around $$$2 \ln n$$$. Now for a given $$$n$$$, can you help Baozii find the exact expected number of nodes?
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). The description of the test cases follows.
The only line of each test case contains an integer $$$n$$$ ($$$1 \le n \le 10^{18}$$$) — the length of the initial sequence.
For each test case, output a real number: the expected number of nodes in the Chtholly Tree. The absolute difference between your answer and the jury's answer should not be more than $$$10^{-3}$$$.
512345
1.000000 1.666667 2.200000 2.642857 3.020202
