Problem:
Score : 400 points

Problem Statement

We say that a odd number N is similar to 2017 when both N and (N+1)/2 are prime.

You are given Q queries.

In the i-th query, given two odd numbers l_i and r_i, find the number of odd numbers x similar to 2017 such that l_i ≤ x ≤ r_i.

Constraints

1≤Q≤10^5

1≤l_i≤r_i≤10^5

l_i and r_i are odd.

All input values are integers.

Input

Input is given from Standard Input in the following format:

Q
l_1 r_1
:
l_Q r_Q

Output

Print Q lines. The i-th line (1≤i≤Q) should contain the response to the i-th query.

Sample Input 1

1
3 7

Sample Output 1

2

3 is similar to 2017, since both 3 and (3+1)/2=2 are prime.

5 is similar to 2017, since both 5 and (5+1)/2=3 are prime.

7 is not similar to 2017, since (7+1)/2=4 is not prime, although 7 is prime.

Thus, the response to the first query should be 2.

Sample Input 2

4
13 13
7 11
7 11
2017 2017

Sample Output 2

1
0
0
1

Note that 2017 is also similar to 2017.

Sample Input 3

6
1 53
13 91
37 55
19 51
73 91
13 49

Sample Output 3

4
4
1
1
1
2

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Linq.Expressions;
using System.Reflection.Emit;
using System.Security.Cryptography;
using System.Text;
namespace Program{
    class MainClass{
        ////////////////////////////////////////////////////////////
        private int Q;
        private int[] l, r;
        void Solve(){
            Q = io.Int;
            io.ini(out l,out r,Q);
            var P=new Prime();
            var ans = new int[Q];
            P.SieveInit(100001);
            //var hoge = P.SieveArr();
            //var hogehoge = hoge.Where(v => v.Odd() && P.isPrime[(v + 1) / 2]).ToArray();           
            Q.REP(i =>{
                for(int j = l[i]; j <= r[i]; j += 2){
                    if(P.isPrime[j]&&P.isPrime[(j+1)/2]) { ans[i]++; }    
                }                
            });
            io.o(ans);
        }             

        ////////////////////////////////////////////////////////////
        public static void Main() { new MainClass().Stream(); }
        IO io = new IO();
        Mat mat = new Mat();
        void Stream(){
            Solve();
            io.writeFlush();
        }
        //void Stream() { Test(); io.writeFlush(); }
        void Test() { }
        #region MockMacro
        //cannot use break,continue,goto
        void FOR(int a, int b, Action<int> act){
            for(int i = a; i < b; ++i) act(i);
        }
        void FORR(int a, int b, Action<int> act){
            for(int i = a - 1; i >= b; --i) act(i);
        }
        #endregion
        ////////////////////////////////////////////////////////////
    }
    class Prime {
        public bool[] isPrime;
        private bool[] _isPrime;
        private long f, t;
        public bool IsPrime(long n) {
            if (n == 2) return true; if (n < 2 || n % 2 == 0) return false;
            for (long v = 3; v <= (long)Math.Sqrt(n); v += 2) if (n % v == 0) return false;
            return true;
        }
        #region sieve
        public void SieveInit(int n) {
            isPrime = new bool[n + 1];
            for (int i = 2; i <= n; ++i)
                isPrime[i] = true;
            for (int i = 2; i <= n; ++i) {
                if (isPrime[i])
                    for (int j = 2 * i; j <= n; j += i)
                        isPrime[j] = false;
            }
        }
        public int[] SieveArr() {
            var li = new List<int>();
            for (int i = 0; i < isPrime.Length; ++i)
                if (isPrime[i])
                    li.Add(i);
            return li.ToArray();
        }
        public int SieveNum() {
            return isPrime.Where(f => f).Count();
        }
        #endregion
        #region segmentSeve
        public void SegmentSieveInit(long a, long b) {
            f = Math.Min(a, b);
            t = Math.Max(a, b);
            isPrime = new bool[t - f];
            _isPrime = new bool[(int)Math.Sqrt(t) + 1];
            for (int i = 0; i < t - f; ++i) isPrime[i] = true;
            for (int i = 0; (long)i * i < t; ++i) _isPrime[i] = true;
            for (int i = 2; (long)i * i < t; ++i) {
                if (_isPrime[i]) {
                    for (int j = 2 * i; (long)j * j < t; j += i)
                        _isPrime[j] = false;
                    for (long j = Math.Max(2L, (f + i - 1) / i) * i; j < t; j += i)
                        isPrime[j - f] = false;
                }
            }
        }
        public long[] SegmentSieveArr() {
            var _prime_num = new List<long>();
            for (int i = 0; i < t - f; ++i)
                if (isPrime[i])
                    _prime_num.Add((long)i + f);
            return _prime_num.ToArray();
        }
        public int SegmentSieveNum() {
            return isPrime.Where(f => f).Count();
        }
        #endregion
    }

    #region default
    class IO{
        string[] nextBuffer;
        int BufferCnt;
        char[] cs = new char[]{' '};
        StreamWriter sw = new StreamWriter(Console.OpenStandardOutput()){AutoFlush = false};
        public IO(){
            nextBuffer = new string[0];
            BufferCnt = 0;
            Console.SetOut(sw);
        }
        public string Next(){
            if(BufferCnt < nextBuffer.Length) return nextBuffer[BufferCnt++];
            string st = Console.ReadLine();
            while(st == "") st = Console.ReadLine();
            nextBuffer = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
            BufferCnt = 0;
            return nextBuffer[BufferCnt++];
        }
        public string String => Next();
        public char Char => char.Parse(String);
        //public int Int => int.Parse(String);
        public int Int => String.toInt();
        public long Long => String.toLong();
        public double Double => double.Parse(String);
        public string[] arr => Console.ReadLine().Split(' ');
        public char[] arrChar => Array.ConvertAll(arr, char.Parse);
        //public int[] arrInt => Array.ConvertAll(arr, int.Parse);
        public int[] arrInt => arr.Select(v => v.toInt()).ToArray();
        //public long[] arrLong => Array.ConvertAll(arr, long.Parse);
        public long[] arrLong => arr.Select(v => v.toLong()).ToArray();
        public double[] arrDouble => Array.ConvertAll(arr, double.Parse);
        public T i<T>() { return suitType<T>(String); }
        public void i<T>(out T v) { v = suitType<T>(String); }
        public void i<T, U>(out T v1, out U v2){
            i(out v1);
            i(out v2);
        }
        public void i<T, U, V>(out T v1, out U v2, out V v3){
            i(out v1);
            i(out v2);
            i(out v3);
        }
        public void i<T, U, V, W>(out T v1, out U v2, out V v3, out W v4){
            i(out v1);
            i(out v2);
            i(out v3);
            i(out v4);
        }
        public void i<T, U, V, W, X>(out T v1, out U v2, out V v3, out W v4, out X v5){
            i(out v1);
            i(out v2);
            i(out v3);
            i(out v4);
            i(out v5);
        }
        public void ini<T>(out T[] a, int n){
            a = new T[n];
            for(int i = 0; i < n; ++i) a[i] = suitType<T>(String);
        }
        public void ini<T, U>(out T[] a, out U[] b, int n){
            a = new T[n];
            b = new U[n];
            for(int i = 0; i < n; ++i){
                a[i] = i<T>();
                b[i] = i<U>();
            }
        }
        public void ini<T, U, V>(out T[] a, out U[] b, out V[] c, int n){
            a = new T[n];
            b = new U[n];
            c = new V[n];
            for(int i = 0; i < n; ++i){
                a[i] = i<T>();
                b[i] = i<U>();
                c[i] = i<V>();
            }
        }
        public void ini<T, U>(out Tuple<T, U>[] t, int n){
            t = new Tuple<T, U>[n];
            for(int j = 0; j < n; ++j){
                T i1;
                U i2;
                i(out i1);
                i(out i2);
                t[j] = Tuple.Create(i1, i2);
            }
        }
        public void ini<T, U, V>(out Tuple<T, U, V>[] t, int n){
            t = new Tuple<T, U, V>[n];
            for(int j = 0; j < n; ++j){
                T i1;
                U i2;
                V i3;
                i(out i1);
                i(out i2);
                i(out i3);
                t[j] = Tuple.Create(i1, i2, i3);
            }
        }
        public void ini<T>(out T[,] a, int h, int w){
            a = new T[h, w];
            for(int i = 0; i < h; ++i) for(int j = 0; j < w; ++j) a[i, j] = i<T>();
        }
        public void o<T>(T v) { Console.WriteLine(v); }
        public void o<T>(params T[] a) { Array.ForEach(a, n => o(n)); }
        public void o<T>(T[,] a){
            int a0 = a.GetLength(0), a1 = a.GetLength(1);
            for(int i = 0; i < a0; ++i){
                for(int j = 0; j < a1 - 1; ++j) Console.Write(a[i, j] + " ");
                Console.WriteLine(a[i, a1 - 1]);
            }
        }
        public void YN(bool f) { o(f ? "YES" : "NO"); }
        public void Yn(bool f) { o(f ? "Yes" : "No"); }
        public void yn(bool f) { o(f ? "yes" : "no"); }
        public void ol<T>(params T[] a) { o(connect<T>(a)); }
        public void or<T>(T a) { Console.Write(a); }
        public void br() { o(""); }
        public void writeFlush() { Console.Out.Flush(); }
        private string connect<T>(params T[] s) { return string.Join(" ", s); }
        private bool typeEQ<T, U>() { return typeof(T).Equals(typeof(U)); }
        private T convertType<T, U>(U v) { return (T) Convert.ChangeType(v, typeof(T)); }
        private T suitType<T>(string s){
            if(typeEQ<T, int>()) return convertType<T, int>(int.Parse(s));
            if(typeEQ<T, long>()) return convertType<T, long>(long.Parse(s));
            if(typeEQ<T, double>()) return convertType<T, double>(double.Parse(s));
            if(typeEQ<T, char>()) return convertType<T, char>(char.Parse(s));
            return convertType<T, string>(s);
        }
    }
    class PQueue<T> where T : struct, IComparable<T>{
        public List<T> heap;
        private Comparison<T> comp;
        private IComparer<T> comparer;
        private int size;
        private int type;//type=0->min
        public PQueue(int type = 0) : this(Comparer<T>.Default) { this.type = type; }
        public PQueue(IComparer<T> comparer) : this(16, comparer.Compare) { this.comparer = comparer; }
        public PQueue(Comparison<T> comparison) : this(16, comparison) { }
        public PQueue(int capacity, Comparison<T> comparison){
            this.heap = new List<T>(capacity);
            this.comp = comparison;
        }
        public void Enqueue(T item){
            this.heap.Add(item);
            var i = size++;
            while(i > 0){
                var p = (i - 1) >> 1;
                if(Compare(this.heap[p], item) <= 0) break;
                this.heap[i] = heap[p];
                i = p;
            }
            this.heap[i] = item;
        }
        public T Dequeue(){
            var ret = this.heap[0];
            var x = this.heap[--size];
            var i = 0;
            while((i << 1) + 1 < size){
                var a = (i << 1) + 1;
                var b = (i << 1) + 2;
                if(b < size && Compare(heap[b], heap[a]) < 0) a = b;
                if(Compare(heap[a], x) >= 0) break;
                heap[i] = heap[a];
                i = a;
            }
            heap[i] = x;
            heap.RemoveAt(size);
            return ret;
        }
        public T Peek() { return heap[0]; }
        public int Count { get { return size; } }
        public bool Any() { return size > 0; }
        public bool Empty() { return !Any(); }
        public bool Contains(T v) { return heap.Contains(v); }
        private int Compare(T x, T y) { return type == 0 ? x.CompareTo(y) : y.CompareTo(x); }
    }
    class AssociativeArray<T> : IEnumerable{
        public Dictionary<T, int> dic;
        public AssociativeArray() { dic = new Dictionary<T, int>(); }
        public AssociativeArray(params T[] a){
            dic = new Dictionary<T, int>();
            Add(a);
        }
        public void Add(T a){
            if(!conK(a)) dic[a] = 0;
            dic[a]++;
        }
        public void Add(params T[] a){
            a.Length.REP(i =>{
                if(!conK(a[i])) dic[a[i]] = 0;
                dic[a[i]]++;
            });
        }
        public void Set(T k, int v){
            if(!dic.ContainsKey(k)) dic[k] = 0;
            dic[k] = v;
        }
        public void Remove(params T[] a){
            a.Length.REP(i =>{
                if(conK(a[i])) dic.Remove(a[i]);
            });
        }
        public T[] Keys() { return dic.Keys.ToArray<T>(); }
        public int Val(T k) { return (dic.ContainsKey(k)) ? dic[k] : 0; }
        public int Val(params T[] k) { return k.Aggregate(0, (x, xs) => x + Val(xs)); }
        public int ValSum => dic.Values.Sum();
        public int KeyNum => dic.Keys.Count;
        public int MaxVal => dic.Values.Max();
        public int MinVal => dic.Values.Min();
        public T MaxKey => MaxK();
        public T MinKey => MinK();
        public T MaxK(){
            var maxV = MaxVal;
            return dic.First(d => d.Value == maxV).Key;
        }
        public T MinK(){
            var minV = MinVal;
            return dic.First(d => d.Value == minV).Key;
        }
        public T[] MaxKeys(){
            var maxV = MaxVal;
            return dic.Where(kp => kp.Value == maxV).ToDictionary(kp => kp.Key, kp => kp.Value).Keys.ToArray();
        }
        public T[] MinKeys(){
            var minV = MinVal;
            return dic.Where(kp => kp.Value == minV).ToDictionary(kp => kp.Key, kp => kp.Value).Keys.ToArray();
        }
        public bool conK(T k) { return dic.ContainsKey(k); }
        public bool anyK(params T[] k) { return k.Any(key => conK(key)); }
        public bool allK(params T[] k) { return k.All(key => conK(key)); }
        public void Show(){
            foreach(var v in dic) { Console.WriteLine(v.Key + " : " + v.Value); }
        }
        public T[] ValueSortedKey(bool inc = true) { return (inc ? (dic.OrderBy(kp => kp.Value)) : (dic.OrderByDescending(kp => kp.Value))).ThenBy(kp => kp.Key).ToDictionary(kp => kp.Key, kp => kp.Value).Keys.ToArray(); }
        public IEnumerator GetEnumerator(){
            foreach(var kp in dic) yield return kp;
        }
        //:sort->array
    }
    public struct edge : IComparable<edge>{
        public int to;
        public long cost;
        public edge(int to, long cost){
            this.to = to;
            this.cost = cost;
        }
        public int CompareTo(edge e) { return cost.CompareTo(e.cost); }
    }
    public struct Indexed : IComparable<Indexed>{
        public long Value;
        public int Index;
        public Indexed(long value, int index){
            Value = value;
            Index = index;
        }
        public int CompareTo(Indexed i) { return Value.CompareTo(i.Value); }
    }
    class Mat{
        public long mod = 1000000007;//10^9+7
        public long Pow(long a, long b){
            if(b == 0) return 1;
            if(b % 2 == 1) return (a % mod * Pow(a % mod, b - 1) % mod) % mod;
            else return Pow(a * a % mod, b / 2) % mod;
        }
        public long Fact(long n){
            long ret = 1;
            for(long i = 1; i <= n; i++) ret = (ret * i) % mod;
            return ret;
        }
        public long ModC(long n, long r){
            if(r == 0 || n == r) return 1;
            if(n == 0) return 0;
            if(n < 0 || n < r) throw new ArgumentException("n,r invalid");
            else return (Fact(n) % mod * Pow((Fact(n - r) % mod * Fact(r) % mod) % mod, mod - 2) % mod) % mod;
        }
        public long[,] C(int N){
            long[,] Co = new long[N + 1, N + 1];
            (N + 1).REP(i => (i + 1).REP(j => Co[i, j] = (j == 0 || j == i) ? 1L : Co[i - 1, j - 1] + Co[i - 1, j]));
            return Co;
        }
        public long DupC(long n, long r) { return ModC(n + r - 1, r); }
        public long P(long n, long r) { return Fact(n) / (Fact(n - r)); }//test
        public bool isPrime(long n){
            if(n == 2) return true;
            if(n < 2 || n % 2 == 0) return false;
            for(long v = 3; v <= (long) Math.Sqrt(n); v += 2) if(n % v == 0) return false;
            return true;
        }
        public long LCM(long a, long b) { return a * (b / GCD(a, b)); }
        public long LCM(params long[] a) { return a.Aggregate((v, n) => LCM(v, n)); }
        public long GCD(long a, long b){
            if(a < b) Swap(ref a, ref b);
            return b == 0 ? a : GCD(b, a % b);
        }
        public long GCD(params long[] array) { return array.Aggregate((v, n) => GCD(v, n)); }
        public T Max<T>(params T[] a) { return a.Max(); }
        public T Min<T>(params T[] a) { return a.Min(); }
        public void Swap<T>(ref T a, ref T b){
            T tmp = a;
            a = b;
            b = tmp;
        }
        public double Dis(int x1, int y1, int x2, int y2) { return Math.Sqrt(Math.Pow((x2 - x1), 2) + Math.Pow((y2 - y1), 2)); }
        public int mDis(int x1, int y1, int x2, int y2) { return Math.Abs(x1 - x2) + Math.Abs(y1 - y2); }
        public int[] DigArr(int n){
            int[] ret = new int[Digit(n)];
            ret.Length.REP(i => ret[i] = DigVal(n, i + 1));
            return ret;
        }
        public long DigArr2Num(IEnumerable<int> enu) { return enu.Aggregate((v, n) => v * 10 + n); }
        public int Digit(long n) { return (n == 0) ? 1 : (int) Math.Log10(n) + 1; }
        public int DigVal(int n, int dig) { return (n % (int) Pow(10, dig)) / (int) Pow(10, dig - 1); }
        public long Tousa(long a, long d, long n) { return a + (n - 1) * d; }
        public long TousaSum(long a, long d, long n) { return n * (2 * a + (n - 1) * d) / 2; }
        public long[] EnuDivsor(long N){
            var ret = new SortedSet<long>();
            for(long i = 1; i * i <= N; i++)
                if(N % i == 0){
                    ret.Add(i);
                    ret.Add(N / i);
                }
            return ret.ToArray();
        }
        public Dictionary<long, long> PrimeFactor(long n){
            var ret = new Dictionary<long, long>();
            for(int i = 2; i * i <= n; ++i){
                if(!ret.ContainsKey(i)) ret[i] = 0;
                while(n % i == 0){
                    ++ret[i];
                    n /= i;
                }
            }
            return ret.Where(kp => kp.Value != 0).ToDictionary(kp => kp.Key, kp => kp.Value);
        }
    }
    #endregion
    #region Ex
    static class StringEX{
        public static string Reversed(this string s) { return string.Join("", s.Reverse()); }
        public static string Repeat(this string s, int n) { return string.Concat(Enumerable.Repeat(s, n).ToArray()); }
        public static int toInt(this string s){
            int n;
            return (int.TryParse(s.TrimStart('0'), out n)) ? n : 0;
        }
        public static int toInt(this char c) { return toInt(c.ToString()); }
        public static int toInt(this char[] c) { return toInt(new string(c)); }
        public static long[] toLong(this int[] a) { return a.Select(v => (long) v).ToArray(); }
        public static long[,] toLong(this int[,] m){
            var h = m.GetLength(0);
            var w = m.GetLength(1);
            var ret = new long[h, w];
            for(int i = 0; i < h; ++i) for(int j = 0; j < w; ++j) ret[i, j] = (long) m[i, j];
            return ret;
        }
        public static long toLong(this string s){
            long n;
            return (long.TryParse(s.TrimStart('0'), out n)) ? n : (long) 0;
        }
        public static long toLong(this char c) { return toLong(c.ToString()); }
        public static long toLong(this char[] c) { return toLong(new string(c)); }
        public static string toString(this char[] c) { return new string(c); }
    }
    static class NumericEx{
        public static string pad0<T>(this T v, int n) { return v.ToString().PadLeft(n, '0'); }
        public static double RoundOff(this double v, int n) { return Math.Round(v, n - 1, MidpointRounding.AwayFromZero); }
        public static double RoundUp(this double v) { return Math.Ceiling(v); }
        public static double RoundDown(this double v) { return Math.Floor(v); }
        public static bool Odd(this int v) { return v % 2 != 0; }
        public static bool Odd(this long v) { return v % 2 != 0; }
        public static void REP(this int v, Action<int> act){
            for(int i = 0; i < v; ++i) act(i);
        }
        public static void REPR(this int v, Action<int> act){
            for(int i = v - 1; i >= 0; --i) act(i);
        }
    }
    static class ArrayEX{
        public static T[] Sort<T>(this T[] a){
            Array.Sort(a);
            return a;
        }
        public static T[] SortR<T>(this T[] a){
            Array.Sort(a);
            Array.Reverse(a);
            return a;
        }
        public static void Set<T>(this T[] a, T v) { a.Length.REP(i => a[i] = v); }
        public static void Set<T>(this T[,] a, T v) { a.GetLength(0).REP(i => a.GetLength(1).REP(j => a[i, j] = v)); }
        public static int[] toIntArr(this string a) { return a.Select(c => c.toInt()).ToArray(); }
        public static int[] toIntArr(this string[] a) { return Array.ConvertAll(a, int.Parse); }
        public static long[] toLongArr(this string a) { return a.Select(c => c.toLong()).ToArray(); }
        public static long[] toLongArr(this string[] a) { return Array.ConvertAll(a, long.Parse); }
        public static double[] toDoubleArr(this string[] a) { return Array.ConvertAll(a, double.Parse); }
        public static char[] toCharArr(this string[] a) { return Array.ConvertAll(a, char.Parse); }
        public static int LowerBound<T>(this T[] a, T x) where T : IComparable{
            Array.Sort(a);
            int lb = -1, ub = a.Length;
            while(ub - lb > 1){
                int mid = (ub + lb) / 2;
                if(a[mid].CompareTo(x) >= 0) ub = mid;
                else lb = mid;
            }
            return ub;
        }
        public static int UpperBound<T>(this T[] a, T x) where T : IComparable{
            Array.Sort(a);
            int lb = -1, ub = a.Length;
            while(ub - lb > 1){
                int mid = (ub + lb) / 2;
                if(a[mid].CompareTo(x) > 0) ub = mid;
                else lb = mid;
            }
            return ub;
        }
    }
    #endregion
}