Buggy python submission (verdict: Time Limit Exceeded):
def main():
    import sys
    from fractions import gcd
    input = sys.stdin.readline

    def init(init_val):
        #set_val
        for i in range(n):
            seg[i+num-1]=init_val[i]    
        #built
        for i in range(num-2,-1,-1) :
            seg[i]=gcd(seg[2*i+1],seg[2*i+2]) 
        
    def update(k,x):
        k += num-1
        seg[k] = x
        while k:
            k = (k-1)//2
            seg[k] = gcd(seg[k*2+1],seg[k*2+2])
        
    def query(p,q):
        if q<=p:
            return ide_ele
        p += num-1
        q += num-2
        res=ide_ele
        while q-p>1:
            if p&1 == 0:
                res = gcd(res,seg[p])
            if q&1 == 1:
                res = gcd(res,seg[q])
                q -= 1
            p = p//2
            q = (q-1)//2
        if p == q:
            res = gcd(res,seg[p])
        else:
            res = gcd(gcd(res,seg[p]),seg[q])
        return res



    n = int(input())
    a = tuple(map(int,input().split()))

    #####単位元######
    ide_ele = 0

    #num:n以上の最小の2のべき乗
    num =2**(n-1).bit_length()
    seg=[ide_ele]*2*num
    init(a)

    ans = -1
    for i in range(n):
        ans = max(ans,gcd(query(0,i),query(i+1,n)))
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
N = int(input())
A = list(map(int,input().split()))

X = [0]*N
Y = [0]*N

import fractions

X[0]=A[0]
for i in range(1,N):
    X[i] = fractions.gcd(X[i-1], A[i])

Y[N-1]=A[N-1]
for i in range(1,N):
    Y[N-1-i] = fractions.gcd(Y[N-i], A[N-1-i])

cand = []
cand.append(X[N-2])
cand.append(X[N-1])#=Y[0]
cand.append(Y[1])
for i in range(1,N-1):
    cand.append(fractions.gcd(X[i-1],Y[i+1]))

print(max(cand))

Buggy python submission (verdict: Time Limit Exceeded):
import fractions
#####segfunc######
def segfunc(x,y):
    return fractions.gcd(x,y)

class SegmentTree():
    def __init__(self,A,n,ide_ele):
        self.ide_ele=ide_ele
        self.num =2**(n-1).bit_length()
        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]=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] = 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 = segfunc(res,self.seg[p])
            if q&1 == 1:
                res = segfunc(res,self.seg[q])
                q -= 1
            p = p//2
            q = (q-1)//2
        if p == q:
            res = segfunc(res,self.seg[p])
        else:
            res = segfunc(segfunc(res,self.seg[p]),self.seg[q])
        return res

N = int(input())
A = list(map(int,input().split()))
ST = SegmentTree(A,N,0)
out = 1
for i in range(N):
    X = ST.query(0,i)
    Y = ST.query(i+1,N)
    out = max(out,fractions.gcd(X,Y))
print(out)

Fixed submission:
N = int(input())
A = list(map(int,input().split()))

X = [0]*N
Y = [0]*N

import fractions

X[0]=A[0]
for i in range(1,N):
    X[i] = fractions.gcd(X[i-1], A[i])

Y[N-1]=A[N-1]
for i in range(1,N):
    Y[N-1-i] = fractions.gcd(Y[N-i], A[N-1-i])

cand = []
cand.append(X[N-2])
cand.append(X[N-1])#=Y[0]
cand.append(Y[1])
for i in range(1,N-1):
    cand.append(fractions.gcd(X[i-1],Y[i+1]))

print(max(cand))