Given a non-empty array of unique positive integers A, consider the following graph:

There are A.length nodes, labelled A[0] to A[A.length - 1];
There is an edge between A[i] and A[j] if and only if A[i] and A[j] share a common factor greater than 1.

Return the size of the largest connected component in the graph.
 



Example 1:
Input: [4,6,15,35]
Output: 4



Example 2:
Input: [20,50,9,63]
Output: 2



Example 3:
Input: [2,3,6,7,4,12,21,39]
Output: 8


Note:

1 <= A.length <= 20000
1 <= A[i] <= 100000

```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

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "from typing import Deque, Dict, List, Set, Tuple\n'''\n952. Largest Component Size by Common Factor.  Hard\n\nGiven a non-empty array of unique positive integers A,\nconsider the following graph:\n\nThere are A.length nodes, labelled A[0] to A[A.length - 1];\nThere is an edge between A[i] and A[j] if and only if\nA[i] and A[j] share a common factor greater than 1.\nReturn the size of the largest connected component in the graph.\n\nExample 1:\nInput: [4,6,15,35]\nOutput: 4\n\nExample 2:\nInput: [20,50,9,63]\nOutput: 2\n\nExample 3:\nInput: [2,3,6,7,4,12,21,39]\nOutput: 8\n\nNote:\n1 <= A.length <= 20000\n1 <= A[i] <= 100000\n\nAccepted 18,093 / 53,552 submissions.\n'''\n\n\ndef gen_primes():\n    '''\n    Prime number generator, optimized to try only odd numbers.\n    How to populate a list with some range of generated values:\n        list(itertools.islice((p for p in gen_primes_opt()), beg_num, end_num)))\n    '''\n    yield 2                                     # return the first prime number\n    prime_divs = collections.defaultdict(list)  # map numbers to their prime divisors\n    candidate = 3\n    while True:\n        if candidate in prime_divs:\n            for prime_div in prime_divs[candidate]:\n                prime_divs[prime_div*2 + candidate].append(prime_div)\n            del prime_divs[candidate]           # sieve no longer needed for candidate\n        else:\n            yield candidate                     # yield the next prime number\n            prime_divs[candidate * candidate] = [candidate] # start sieve for sqaure\n        candidate += 2\n\n\n\nclass Solution:\n\n    def __init__(self):\n        self.prime_gen = gen_primes()\n        self.primes = [next(self.prime_gen) for _ in range(55)]\n\n\n    def get_primes(self, lower_bound: int) -> List[int]:\n        while self.primes[-1] < lower_bound:\n            self.primes.append(next(self.prime_gen))\n        return self.primes\n\n    def prime_fac_set(self, num: int):\n        pfs = set()\n        for prime in self.primes:\n            if num % prime == 0:\n                pfs.add(prime)\n\n            if prime >= num:\n                break\n        return pfs\n\n\n    def largestComponentSizeWip(self, A: List[int]) -> int:\n        return 0\n\n\n        pcomps = collections.defaultdict(set)\n        edges = collections.defaultdict(set)\n        for num in A:\n            pfs = self.prime_fac_set(num)\n            prv = None\n            for prime in pfs:\n                pcomps[prime].add(num)\n                if prv is None:\n                    prv = prime\n                else:\n                    edges[prv].add(prime)\n                    prv = prime\n        max_size = 0\n        print(edges)\n        for prm, num_set in list(pcomps.items()):\n            the_set = num_set\n            the_prm = prm\n            while the_prm in edges:\n                the_set.update(edges[the_prm])\n            max_size = max(max_size, len(the_set))\n\n\n        return max_size\n\n\n\n    def largestComponentSize(self, A: List[int]) -> int:\n        '''\n        Step 1: get primes less than sqrt(max(A))\n        Step 2: compute factors for each number in A\n        Step 3: use union-find to link two primes\n                if they are factors of the same number in A\n        Step 4: for each number in A, add to its union\n                (the union represented by its smallest prime)\n                \n        Runtime: 1000 ms, faster than 99.35% of Python3 online submissions for Largest Component Size by Common Factor.\n        Memory Usage: 19.4 MB, less than 55.56% of Python3 online submissions for Largest Component Size by Common Factor.\n        '''\n        # primes = []\n        # for x in range(2, int(max(A)**0.5)+1):\n        #     for y in primes:\n        #         if x % y == 0:\n        #             break\n        #     else:\n        #         primes.append(x)\n\n        lowerb = int(max(A)**0.5) + 1\n        primes = self.get_primes(lowerb)\n\n        factors = collections.defaultdict(list)         # compute factors of each 'a'\n        for a in A:\n            x = a\n            for p in primes:\n                if p*p > x:\n                    break\n                if x % p == 0:\n                    factors[a].append(p)\n                    while x % p == 0:\n                        x //= p\n            if x > 1:                                   # a new prime found\n                factors[a].append(x)\n                primes.append(x)\n\n        primes = list(set(primes))\n        n = len(primes)\n        p2i = {p: i for i,p in enumerate(primes)}       # prime to index\n\n        parent = [i for i in range(n)]                  # union-find on primes\n\n        def find(i):\n            if i != parent[i]:\n                parent[i] = find(parent[i])\n            return parent[i]\n\n        def union(i,j):\n            pi, pj = find(i), find(j)\n            if pi != pj:\n                parent[pi] = pj\n\n        for a in A:\n            if factors[a]:\n                p0 = factors[a][0]\n                for p in factors[a][1:]:                # link two primes if they are factors of 'a'\n                    union(p2i[p0], p2i[p])\n\n        count = collections.Counter(find(p2i[factors[a][0]]) for a in A if factors[a])      # each 'a' corresponds to a prime index\n        return max(count.values())\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'largestComponentSize'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 16344