<p>Given the head of a linked list <code>head</code>, in which each node contains an integer value.</p>

<p>Between every pair of adjacent nodes, insert a new node with a value equal to the <strong>greatest common divisor</strong> of them.</p>

<p>Return <em>the linked list after insertion</em>.</p>

<p>The <strong>greatest common divisor</strong> of two numbers is the largest positive integer that evenly divides both numbers.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/18/ex1_copy.png" style="width: 641px; height: 181px;" />
<pre>
<strong>Input:</strong> head = [18,6,10,3]
<strong>Output:</strong> [18,6,6,2,10,1,3]
<strong>Explanation:</strong> The 1<sup>st</sup> diagram denotes the initial linked list and the 2<sup>nd</sup> diagram denotes the linked list after inserting the new nodes (nodes in blue are the inserted nodes).
- We insert the greatest common divisor of 18 and 6 = 6 between the 1<sup>st</sup> and the 2<sup>nd</sup> nodes.
- We insert the greatest common divisor of 6 and 10 = 2 between the 2<sup>nd</sup> and the 3<sup>rd</sup> nodes.
- We insert the greatest common divisor of 10 and 3 = 1 between the 3<sup>rd</sup> and the 4<sup>th</sup> nodes.
There are no more adjacent nodes, so we return the linked list.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/07/18/ex2_copy1.png" style="width: 51px; height: 191px;" />
<pre>
<strong>Input:</strong> head = [7]
<strong>Output:</strong> [7]
<strong>Explanation:</strong> The 1<sup>st</sup> diagram denotes the initial linked list and the 2<sup>nd</sup> diagram denotes the linked list after inserting the new nodes.
There are no pairs of adjacent nodes, so we return the initial linked list.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the list is in the range <code>[1, 5000]</code>.</li>
	<li><code>1 &lt;= Node.val &lt;= 1000</code></li>
</ul>

# Definition for singly-linked list.
# class ListNode:
#     def __init__(self, val=0, next=None):
#         self.val = val
#         self.next = next
class Solution:
    def insertGreatestCommonDivisors(self, head: Optional[ListNode]) -> Optional[ListNode]:
        global primes, primeList
        primes = set()
        primeList = []
        arr = [1] * 5001

        def seive():
            global primes, primeList
            for i in range(3, 5001, 2):
                # if i * i > 5000:
                #     return
                if arr[i] == 0:
                    continue
                for j in range(i * i, 5001, 2 * i):
                    if j > 5000:
                        break
                    arr[j] = 0
            # print(arr)
            for i in range(3, 5000, 2):
                if arr[i] == 1:
                    primes.add(i)
            primes.add(2)
            primeList += sorted(list(primes))

        def getPrimeFactors(n):
            global primes, primeList
            factor = 0
            factors = {}
            # print(primeList, primes)
            while n > 1:
                if n % primeList[factor] == 0:
                    n //= primeList[factor]
                    if primeList[factor] in factors:
                        factors[primeList[factor]] += 1
                    else:
                        factors[primeList[factor]] = 1
                else:
                    factor += 1
            return factors

        def getGCD(curr, next):
            res = 1
            # print(curr)
            for k, v in curr.items():
                if k in next:
                    # print(k)
                    res *= k ** min(v, next[k])
            return res

        primeFactors = {}
        seive()

        n = head
        curr = getPrimeFactors(head.val)
        primeFactors[head.val] = curr
        while n.next:
            if n.next.val in primeFactors:
                next = primeFactors[n.next.val]
            else:
                next = getPrimeFactors(n.next.val)
                primeFactors[n.next.val] = next
            val = getGCD(curr, next)
            # print(curr, next, val)
            newNode = ListNode(val, n.next)
            n.next = newNode
            n = n.next.next
            curr = next
        return head
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