Buggy python submission (verdict: Time Limit Exceeded):
class SegmentTree():
    def segfunc(self, x, y):
        from fractions import gcd
        if x > y:
            x, y = y, x
        if x == -1:
            return y
        return gcd(x, y) #関数
    
    def __init__(self, a): #a:リスト
        n = len(a)
        self.ide_ele = -1 #単位元
        self.num = 2**((n-1).bit_length()) #num:n以上の最小の2のべき乗
        self.seg = [self.ide_ele]*2*self.num
        
        #set_val
        for i in range(n):
            self.seg[i+self.num-1] = a[i]
        #built
        for i in range(self.num-2,-1,-1):
            self.seg[i] = self.segfunc(self.seg[2*i+1], self.seg[2*i+2])
        
        
    def update(self, k, x):
        k += self.num-1
        self.seg[k] = x
        while k:
            k = (k-1)//2
            self.seg[k] = self.segfunc(self.seg[k*2+1], self.seg[k*2+2])
            
    def query(self, p, q):
        if q <= p:
            return self.ide_ele
        p += self.num-1
        q += self.num-2
        res = self.ide_ele
        while q - p > 1:
            if p&1 == 0:
                res = self.segfunc(res, self.seg[p])
            if q&1 == 1:
                res = self.segfunc(res, self.seg[q])
                q -= 1
            p = p//2
            q = (q-1)//2
        if p == q:
            res = self.segfunc(res,self.seg[p])
        else:
            res = self.segfunc(self.segfunc(res,self.seg[p]),self.seg[q])
        return res
    
N = int(input())
A = list(map(int, input().split()))
M = 0
for i in range(N):
    S = SegmentTree(A)
    S.update(i, -1)
    M = max(M, S.query(0,N))
print(M)

Fixed submission:
class SegmentTree():
    def segfunc(self, x, y):
        if x > y:
            x, y = y, x
        if x == -1:
            return y
        return gcd(x, y) #関数
    
    def __init__(self, a): #a:リスト
        n = len(a)
        self.ide_ele = -1 #単位元
        self.num = 2**((n-1).bit_length()) #num:n以上の最小の2のべき乗
        self.seg = [self.ide_ele]*2*self.num
        
        #set_val
        for i in range(n):
            self.seg[i+self.num-1] = a[i]
        #built
        for i in range(self.num-2,-1,-1):
            self.seg[i] = self.segfunc(self.seg[2*i+1], self.seg[2*i+2])
        
        
    def update(self, k, x):
        k += self.num-1
        self.seg[k] = x
        while k:
            k = (k-1)//2
            self.seg[k] = self.segfunc(self.seg[k*2+1], self.seg[k*2+2])
            
    def query(self, p, q):
        if q <= p:
            return self.ide_ele
        p += self.num-1
        q += self.num-2
        res = self.ide_ele
        while q - p > 1:
            if p&1 == 0:
                res = self.segfunc(res, self.seg[p])
            if q&1 == 1:
                res = self.segfunc(res, self.seg[q])
                q -= 1
            p = p//2
            q = (q-1)//2
        if p == q:
            res = self.segfunc(res,self.seg[p])
        else:
            res = self.segfunc(self.segfunc(res,self.seg[p]),self.seg[q])
        return res
    
from fractions import gcd
    
N = int(input())
A = list(map(int, input().split()))
M = 0
S = SegmentTree(A)
for i in range(N):
    M = max(M, S.segfunc(S.query(0,i), S.query(i+1, N)))
print(M)