Buggy python submission (verdict: Time Limit Exceeded):
#http://tjkendev.github.io/procon-library/python/range_query/rmq_segment_tree.html
from fractions import *
from math import *
import sys
input = sys.stdin.readline

# a_k の値を x に更新
def update(k, x):
    k += N0-1
    data[k] = x
    while k >= 0:
        k = (k - 1) // 2
        #data[k] = min(data[2*k+1], data[2*k+2])
        data[k] = gcd(data[2 * k + 1], data[2 * k + 2])

def check(k, x):
    k += N0-1
    data2 = data[:]
    data2[k] = x
    while k >= 0:
        k = (k - 1) // 2
        data2[k] = gcd(data2[2 * k + 1], data2[2 * k + 2])
    #print(data2)
    return data2[0]

N = int(input())
a = [int(i) for i in input().split()]

# N: 処理する区間の長さ
N0 = 2**(N-1).bit_length()
INF = a[0]
data = [INF]*(2*N0)
#print(data)
for i in range(N):
    update(i, a[i])

#元のセグ木
ans = 1
for i in range(N-1):
    ans = max(ans, check(i+1, a[i]))
else:
    if N % 2 == 1:
        update(N, a[1])
    ans = max(ans, check(0, a[1]))

print(ans)
#print(data)

Fixed submission:
from fractions import *
from math import *

def main():
    n = int(input())
    a = [int(i) for i in input().split()]
    gcd_left = [a[0]]
    gcd_right = [a[-1]]
    for i in range(n-1):
        gcd_left += [gcd(gcd_left[-1], a[i+1])]
        gcd_right += [gcd(gcd_right[-1], a[-i-2])]
    gcd_right = gcd_right[::-1]
    ans = 1
    for i in range(n):
        if i == 0:
            ans = max(ans, gcd_right[i+1])
        elif i == n-1:
            ans = max(ans, gcd_left[i-1])
        else:
            ans = max(ans, gcd(gcd_left[i-1], gcd_right[i+1]))

    print(ans)

main()

Buggy python submission (verdict: Time Limit Exceeded):
#http://tjkendev.github.io/procon-library/python/range_query/rmq_segment_tree.html
from fractions import *
from math import *
import sys
input = sys.stdin.readline

# a_k の値を x に更新
def update(k, x):
    k += N0-1
    data[k] = x
    while k >= 0:
        k = (k - 1) // 2
        #data[k] = min(data[2*k+1], data[2*k+2])
        data[k] = gcd(data[2 * k + 1], data[2 * k + 2])

def check(k, x):
    k += N0-1
    data2 = data[:]
    data2[k] = x
    while k >= 0:
        k = (k - 1) // 2
        data2[k] = gcd(data2[2 * k + 1], data2[2 * k + 2])
    #print(data2)
    return data2[0]

N = int(input())
a = [int(i) for i in input().split()]

# N: 処理する区間の長さ
N0 = 2**(N-1).bit_length()
INF = a[0]
data = [INF]*(2*N0)
#print(data)
for i in range(N):
    update(i, a[i])

#元のセグ木
ans = 1
for i in range(N-1):
    ans = max(ans, check(i+1, a[i]))
else:
    if N % 2 == 1:
        update(N, a[1])
    ans = max(ans, check(0, a[1]))

print(ans)
#print(data)

Fixed submission:
from fractions import *
from math import *

def main():
    n = int(input())
    a = [int(i) for i in input().split()]
    gcd_left = [a[0]]
    gcd_right = [a[-1]]
    for i in range(n-1):
        gcd_left += [gcd(gcd_left[-1], a[i+1])]
        gcd_right += [gcd(gcd_right[-1], a[-i-2])]
    gcd_right = gcd_right[::-1]
    ans = 1
    for i in range(n):
        if i == 0:
            ans = max(ans, gcd_right[i+1])
        elif i == n-1:
            ans = max(ans, gcd_left[i-1])
        else:
            ans = max(ans, gcd(gcd_left[i-1], gcd_right[i+1]))

    print(ans)

main()