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 = 'from math import gcd\nfrom itertools import groupby\n\ndef gcd_split(seq):\n gcds= [int(gcd(a,b)==1) for a,b in zip(seq[1:],seq[:-1])]\n gcds.append(int(gcd(seq[0],seq[-1])==1))\n # print(gcds)\n if max(gcds)==0:\n  return -1\n else:\n  splits=[len(list(x))+1 for num,x in groupby(gcds) if num==0]\n  # print(splits)\n  if gcds[0]==gcds[-1]==0:\n   splits[0] += splits[-1]-1\n   splits = splits[:-1]\n  return splits\n\nfor _ in range(int(input())):\n N=int(input())\n A=[int(x) for x in input().split()]\n split = gcd_split(A)\n # print(split)\n \n res=[]\n if split!=-1:\n  for K in range(2,N+1):\n   temp=(x for x in split if x>=K)\n   ins = sum([(x//(K-1)-1 if x%(K-1)==0 else x//(K-1)) for x in temp])\n   if ins==0:\n    break\n   else:\n    res.append(ins)\n else:\n  for K in range(2,N+1):\n   ins = N//(K-1)+(N%(K-1)>0)\n   if ins==0:\n    break\n   else:\n    res.append(ins)\n \n res = res + [0]*(N-1 -len(res))\n print(*res)'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 13487