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 DBG = false
local function dbgpr(...)
    if DBG then
        io.write("[dbg]")
        print(...)
    end
end

local function dbgpr_t(tbl, use_pairs)
    if DBG then
        local enum = ipairs
        if use_pairs then
            enum = pairs
        end
        dbgpr(tbl)
        io.write("[dbg]")
        for i,v in enum(tbl) do
            io.write(i)
            io.write(":")
            io.write(tostring(v))
            io.write(" ")
        end
        print("")
    end
end

-- SegTree
-- 参考にしました: http://tsutaj.hatenablog.com/entry/2017/03/29/204841
local SegTree = {}
function SegTree.new(func, ident, vec)
    -- func: binary function
    -- ident: identity element
    -- vec: table
    local self = setmetatable({}, {__index = SegTree})
    self.func = func
    self.ident = ident
    local a = {}
    self.a = a
    local n = 1
    while n < #vec do
        n = n * 2
    end
    -- n is smallest power of 2 that is greater than or equal to #vec
    self.n = n
    local size = 2*n - 1
    for i=1, size do
        a[i] = ident
    end
    -- for verification
    --[[
    for i=1, #vec do
        a[i+n-1] = vec[i]
    end
    for i=n-1,1,-1 do
        a[i] = func(a[2*i], a[2*i+1])
    end
    ]]
    return self
end

function SegTree:update(idx, val)
    local a = self.a
    local func = self.func
    idx = idx + self.n - 1
    a[idx] = val
    while true do
        idx = math.floor(idx / 2)
        a[idx] = func(a[idx*2], a[idx*2 + 1])
        if idx == 1 then
            break
        end
    end
end

function SegTree:query(left, right)
    -- query to [left, right)
    -- **be careful about right half open interval**
    local func = self.func
    local ident = self.ident
    local a = self.a
    local function q(k, l, r)
        if r <= left or l >= right then
            return ident
        end
        if l >= left and r <= right then
            return a[k]
        end
        local h = math.floor((l+r)/2)
        local vl = q(2*k  , l, h)
        local vr = q(2*k+1, h, r)
        return func(vl, vr)
    end
    return q(1, 1, self.n + 1)
end
------
local function gcd(a,b)
    if b == 0 then
        return a
    else
        return gcd(b, a % b)
    end
end
----

local N = io.read("*n")
local A = {}
for i=1,N do
    A[i] = io.read("*n")
end
local st = SegTree.new(gcd, 0, A)

-- for verification
for i=1, #st.a do
    st.a[i] = 0
end
for i=1, N do
    st:update(i, A[i])
end

dbgpr_t(st.a)
local maxval = 0
for i=1, N do
    local left = st:query(1, i)
    local right = st:query(i+1, N+1)
    local g = gcd(left, right)
    dbgpr(i, left, right, g)
    maxval = math.max(maxval, g)
end
print(maxval)