Problem:
Score : 600 points

Problem Statement

There is an infinitely large pond, which we consider as a number line.
In this pond, there are N lotuses floating at coordinates 0, 1, 2, ..., N-2 and N-1.
On the lotus at coordinate i, an integer s_i is written.

You are standing on the lotus at coordinate 0. You will play a game that proceeds as follows:

1. Choose positive integers A and B. Your score is initially 0.

2. Let x be your current coordinate, and y = x+A. The lotus at coordinate x disappears, and you move to coordinate y.

If y = N-1, the game ends.

If y \neq N-1 and there is a lotus floating at coordinate y, your score increases by s_y.

If y \neq N-1 and there is no lotus floating at coordinate y, you drown. Your score decreases by 10^{100} points, and the game ends.

3. Let x be your current coordinate, and y = x-B. The lotus at coordinate x disappears, and you move to coordinate y.

If y = N-1, the game ends.

If y \neq N-1 and there is a lotus floating at coordinate y, your score increases by s_y.

If y \neq N-1 and there is no lotus floating at coordinate y, you drown. Your score decreases by 10^{100} points, and the game ends.

4. Go back to step 2.

You want to end the game with as high a score as possible.
What is the score obtained by the optimal choice of A and B?

Constraints

3 \leq N \leq 10^5

-10^9 \leq s_i \leq 10^9

s_0=s_{N-1}=0

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
s_0 s_1 ...... s_{N-1}

Output

Print the score obtained by the optimal choice of A and B.

Sample Input 1

5
0 2 5 1 0

Sample Output 1

3

If you choose A = 3 and B = 2, the game proceeds as follows:

Move to coordinate 0 + 3 = 3. Your score increases by s_3 = 1.

Move to coordinate 3 - 2 = 1. Your score increases by s_1 = 2.

Move to coordinate 1 + 3 = 4. The game ends with a score of 3.

There is no way to end the game with a score of 4 or higher, so the answer is 3. Note that you cannot land the lotus at coordinate 2 without drowning later.

Sample Input 2

6
0 10 -7 -4 -13 0

Sample Output 2

0

The optimal strategy here is to land the final lotus immediately by choosing A = 5 (the value of B does not matter).

Sample Input 3

11
0 -4 0 -99 31 14 -15 -39 43 18 0

Sample Output 3

59

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ProblemA
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var sList = NNList(N);
            if (N == 3)
            {
                Console.WriteLine(0);
                return;
            }
            for (var i = 0; i < N / 2; i++)
            {
                sList[i] += sList[N - i - 1];
            }
            var ans = 0L;
            for (var i = 0; i < N / 2; i++) ans = Max(ans, sList[0] + sList[i]);
            var facts = LIB_Math.Factors(N - 1).Distinct().ToArray();
            foreach (var item in facts.OrderBy(e => e))
            {
                var len = 1L;
                while ((N - 1) % len == 0)
                {
                    var tmp = 0L;
                    for (var i = 0L; i < N / 2; i += len)
                    {
                        tmp += sList[i];
                        ans = Max(ans, tmp);
                    }
                    len *= item;
                }
            }
            Console.WriteLine(ans);
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_Math
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Primes(long x)
        {
            if (x < 2) yield break;
            yield return 2;
            var halfx = x / 2;
            var table = new bool[halfx + 1];
            var max = (long)(Math.Sqrt(x) / 2);
            for (long i = 1; i <= max; ++i)
            {
                if (table[i]) continue;
                var add = 2 * i + 1;
                yield return add;
                for (long j = 2 * i * (i + 1); j <= halfx; j += add)
                    table[j] = true;
            }
            for (long i = max + 1; i <= halfx; ++i)
                if (!table[i] && 2 * i + 1 <= x)
                    yield return 2 * i + 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Factors(long x)
        {
            if (x < 2) yield break;
            while (x % 2 == 0)
            {
                x /= 2; yield return 2;
            }
            var max = (long)Math.Sqrt(x);
            for (long i = 3; i <= max; i += 2)
            {
                while (x % i == 0)
                {
                    x /= i; yield return i;
                }
            }
            if (x != 1) yield return x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Divisor(long x)
        {
            if (x < 1) yield break;
            var max = (long)Math.Sqrt(x);
            for (long i = 1; i <= max; ++i)
            {
                if (x % i != 0) continue;
                yield return i;
                if (i != x / i) yield return x / i;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long GCD(long a, long b)
        {
            while (b > 0)
            {
                var tmp = b;
                b = a % b;
                a = tmp;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long LCM(long a, long b) => a / GCD(a, b) * b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                if (x < long.MaxValue / x) x *= x;
                y >>= 1;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public decimal Sqrt(decimal x)
        {
            decimal prev, cur = (decimal)Math.Sqrt((double)x);
            do
            {
                prev = cur;
                if (prev == 0) return 0;
                cur = (prev + x / prev) / 2;
            } while (cur != prev);
            return cur;
        }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void Build(long n)
        {
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Comb(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Perm(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<List<int>> MakePermutation(long n, bool zeroIndexed = true)
        {
            if (n <= 0) throw new Exception();
            var c = new int[n];
            var a = new int[n];
            if (!zeroIndexed) a[0] = 1;
            for (var i = 1; i < n; i++) a[i] = a[i - 1] + 1;
            yield return new List<int>(a);
            for (var i = 0; i < n;)
            {
                if (c[i] < i)
                {
                    if (i % 2 == 0)
                    {
                        var t = a[0]; a[0] = a[i]; a[i] = t;
                    }
                    else
                    {
                        var t = a[c[i]]; a[c[i]] = a[i]; a[i] = t;
                    }
                    yield return new List<int>(a);
                    ++c[i];
                    i = 0;
                }
                else
                {
                    c[i] = 0;
                    ++i;
                }
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool NextPermutation(long[] ary)
        {
            var n = ary.Length;
            var i = n - 1;
            while (i - 1 >= 0 && ary[i - 1] > ary[i]) --i;
            if (i == 0) return false;
            var j = i;
            while (j + 1 < n && ary[i - 1] < ary[j + 1]) ++j;
            var tmp = ary[i - 1]; ary[i - 1] = ary[j]; ary[j] = tmp;
            var s = i;
            var t = n - 1;
            while (t - s > 0)
            {
                tmp = ary[t]; ary[t] = ary[s]; ary[s] = tmp;
                ++s; --t;
            }
            return true;
        }
    }
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return isEof; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
}