You are given a sequence of integers $A_1, A_2, \ldots, A_N$. This sequence is circular ― for each valid $i$, the element $A_{i+1}$ follows after $A_i$, and the element $A_1$ follows after $A_N$.
You may insert any positive integers at any positions you choose in this sequence; let's denote the resulting sequence by $B$. This sequence is also circular. For each pair of its elements $B_s$ and $B_f$, let's denote the (non-circular) sequence created by starting at $B_s$ and moving from each element to the one that follows after it, until we reach $B_f$, by $B(s, f)$. This sequence includes the elements $B_s$ and $B_f$.
For each $K$ from $2$ to $N$ inclusive, find the smallest possible number of elements that need to be inserted into $A$ to form a sequence $B$ for which there is no subsequence $B(p, q)$ such that:
- The size of $B(p, q)$ is at least $K$.
- There is no pair of consecutive elements in $B(p, q)$ such that their GCD is equal to $1$.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains a single integer $N$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing $N-1$ space-separated integers. For each $i$ ($1 \le i \le N-1$), the $i$-th of these integers should be the smallest number of inserted elements in a valid sequence $B$ for $K = i+1$.

-----Constraints-----
- $1 \le T \le 2,000$
- $2 \le N \le 10^5$
- $1 \le A_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $2 \cdot 10^6$

-----Example Input-----
1
5
3 6 4 5 9

-----Example Output-----
3 1 1 0

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = '# cook your dish here\ndef gcd(m,n):\n if m > n :\n  (a,b) = (n,m)\n else:\n  (a,b) = (m,n)\n r = b\n while r > 1:\n  r = b%a \n  (a,b) = (r,a)\n if r == 0:\n  return b\n else:\n  return 1\n\ndef rlmaker(rl,pc,n):\n d = 1 \n for i in range(0,pc+1):\n  if i <n-1 :\n   rl[i] += pc//d\n   d+=1\n return rl\ndef rlmakers(rl,pc,n):\n d = 1 \n for i in range(0,pc+1):\n  if i <n-1 :\n   if pc%d == 0:\n    rl[i] += pc//d\n   else:\n    rl[i] += ((pc//d)+1)\n   d+=1\n return rl\n  \n  \ndef insFinder(n,l):\n lstone = 0\n rl = []\n for i in range(0,n-1):\n  rl.append(0)\n pathli = []\n pc = 0\n start = -1\n end = -1\n c = 0\n for i in range (0,n-1):\n  if gcd(l[i],l[i+1]) != 1 :\n   if i == 0:\n    start = 0\n   if i == n-2:\n    end = n-1\n   if lstone == 1:\n    pc += 1 \n   else:\n    lstone = 1 \n    pc += 1\n  else:\n   if lstone == 1:\n    pathli.append(pc)\n    pc = 0\n    lstone = 0\n    \n if start == 0:\n  if end == n-1:\n   if gcd(l[n-1],l[0]) != 1:\n    pathli.append(pc)\n    if len(pathli) > 1:\n     pathli[0]+= (pathli[len(pathli)-1]+1)\n     pathli.remove(pathli[len(pathli)-1])\n    else:\n     c = 1\n     pathli[0]+=1\n   else:\n    pathli.append(pc)\n  else:\n   if gcd(l[n-1],l[0]) != 1:\n    pathli[0] += 1\n else:\n  if end == n-1:\n   if gcd(l[n-1],l[0]) != 1:\n    pc += 1\n    pathli.append(pc)\n   else:\n    pathli.append(pc)\n  else:\n   if gcd(l[n-1],l[0]) != 1:\n    pathli.append(1) \n if c == 1:\n  rl = rlmakers(rl,pathli[0],n)\n else:\n  for i in range(0,len(pathli)):\n   rl = rlmaker(rl,pathli[i],n)\n \n for i in range(0,n-1):\n  print(rl[i],end = \' \')\n  \n print("") \n \nnt = int(input())\nfor i in range(0,nt):\n n = int(input())\n l = list(map(int,input().strip().split(\' \')))\n insFinder(n,l)\n \n '
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 13641