Problem:
Score : 300 points

Problem Statement

There are N integers, A_1, A_2, ..., A_N, written on the blackboard.

You will choose one of them and replace it with an integer of your choice between 1 and 10^9 (inclusive), possibly the same as the integer originally written.

Find the maximum possible greatest common divisor of the N integers on the blackboard after your move.

Constraints

All values in input are integers.

2 \leq N \leq 10^5

1 \leq A_i \leq 10^9

Output

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the maximum possible greatest common divisor of the N integers on the blackboard after your move.

Sample Input 1

3
7 6 8

Sample Output 1

2

If we replace 7 with 4, the greatest common divisor of the three integers on the blackboard will be 2, which is the maximum possible value.

Sample Input 2

3
12 15 18

Sample Output 2

6

Sample Input 3

2
1000000000 1000000000

Sample Output 3

1000000000

We can replace an integer with itself.

Solution in LuaJIT (2.0.4) (judge verdict: Accepted)

local ior = io.read
local n = ior("*n")
local a = {}
for i = 1, n do a[i] = io.read("*n") end

local function getgcd(x, y)
  while(0 < x and 0 < y) do
    if(x < y) then y = y % x else x = x % y end
  end
  return math.max(x, y)
end

local mac = math.ceil
local mam = math.max

local mesh = 500

local groups = mac(n / mesh)
local groupgcd = {}
local group_inv_gcd = {}
for i = 1, groups do
  groupgcd[i] = 0
  group_inv_gcd[i] = 0
end

for i = 1, n do
  local g = mac(i / mesh)
  groupgcd[g] = getgcd(groupgcd[g], a[i])
end

for i = 1, groups do
  for j = 1, groups do
    if(i ~= j) then
      group_inv_gcd[i] = getgcd(group_inv_gcd[i], groupgcd[j])
    end
  end
end

local ret = 0

for i = 1, n do
  local g = mac(i / mesh)
  local stt = 1 + (g - 1) * mesh
  local last = (g == groups and n or g * mesh)
  local cand = group_inv_gcd[g]
  for j = stt, last do
    if(j ~= i) then
      cand = getgcd(cand, a[j])
    end
  end
  ret = mam(ret, cand)
end
print(ret)