Querying maximum number of divisors that a number in a given range has

Given Q queries, of type: L R, for each query you must print the maximum number of divisors that a number x (L <= x <= R) has. 
Examples:

L = 1 R = 10:
    1 has 1 divisor.
    2 has 2 divisors.
    3 has 2 divisors.
    4 has 3 divisors.
    5 has 2 divisors.
    6 has 4 divisors.
    7 has 2 divisors.
    8 has 4 divisors.
    9 has 3 divisors.
    10 has 4 divisors.

So the answer for above query is 4, as it is the maximum number of 
divisors a number has in [1, 10].

Pre-requisites : Eratosthenes Sieve, Segment Tree
Below are steps to solve the problem.

1. Firstly, let’s see how many number of divisors does a number n = p₁^k₁ * p₂^k₂ * … * p_n^k_n (where p₁, p₂, …, p_n are prime numbers) has; the answer is (k₁ + 1)*(k₂ + 1)*…*(k_n + 1). How? For each prime number in the prime factorization, we can have its k_i + 1 possible powers in a divisor (0, 1, 2,…, k_i).
2. Now let’s see how can we find the prime factorization of a number, we firstly build an array, smallest_prime[], which stores the smallest prime divisor of i at i^th index, we divide a number by its smallest prime divisor to obtain a new number (we also have the smallest prime divisor of this new number stored), we keep doing it until the smallest prime of the number changes, when the smallest prime factor of the new number is different from the previous number’s, we have k_i for the i^th prime number in the prime factorization of the given number.
3. Finally, we obtain the number of divisors for all the numbers and store these in a segment tree that maintains the maximum numbers in the segments. We respond to each query by querying the segment tree.

C++

// A C++ implementation of the above idea to process
// queries of finding a number with maximum divisors.
#include <bits/stdc++.h>
using namespace std;

#define maxn 1000005
#define INF 99999999

int smallest_prime[maxn];
int divisors[maxn];
int segmentTree[4 * maxn];

// Finds smallest prime factor of all numbers in
// range[1, maxn) and stores them in smallest_prime[],
// smallest_prime[i] should contain the smallest prime
// that divides i
void findSmallestPrimeFactors()
{
    // Initialize the smallest_prime factors of all
    // to infinity
    for (int i = 0 ; i < maxn ; i ++ )
        smallest_prime[i] = INF;

    // to be built like eratosthenes sieve
    for (long long i = 2; i < maxn; i++)
    {
        if (smallest_prime[i] == INF)
        {
            // prime number will have its smallest_prime
            // equal to itself
            smallest_prime[i] = i;
            for (long long j = i * i; j < maxn; j += i)

                // if 'i' is the first prime number reaching 'j'
                if (smallest_prime[j] > i)
                    smallest_prime[j] = i;
        }
    }
}

// number of divisors of n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
// are equal to (k1+1) * (k2+1) ... (kn+1)
// this function finds the number of divisors of all numbers
// in range [1, maxn) and stores it in divisors[]
// divisors[i] stores the number of divisors i has
void buildDivisorsArray()
{
    for (int i = 1; i < maxn; i++)
    {
        divisors[i] = 1;
        int n = i, p = smallest_prime[i], k = 0;

        // we can obtain the prime factorization of the number n
        // n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn) using the
        // smallest_prime[] array, we keep dividing n by its
        // smallest_prime until it becomes 1, whilst we check
        // if we have need to set k zero
        while (n > 1)
        {
            n = n / p;
            k ++;

            if (smallest_prime[n] != p)
            {
                //use p^k, initialize k to 0
                divisors[i] = divisors[i] * (k + 1);
                k = 0;
            }

            p = smallest_prime[n];
        }
    }
}

// builds segment tree for divisors[] array
void buildSegtmentTree(int node, int a, int b)
{
    // leaf node
    if (a == b)
    {
        segmentTree[node] = divisors[a];
        return ;
    }

    //build left and right subtree
    buildSegtmentTree(2 * node, a, (a + b) / 2);
    buildSegtmentTree(2 * node + 1, ((a + b) / 2) + 1, b);

    //combine the information from left
    //and right subtree at current node
    segmentTree[node] = max(segmentTree[2 * node],
                            segmentTree[2 *node + 1]);
}

//returns the maximum number of divisors in [l, r]
int query(int node, int a, int b, int l, int r)
{
    // If current node's range is disjoint with query range
    if (l > b || a > r)
        return -1;

    // If the current node stores information for the range
    // that is completely inside the query range
    if (a >= l && b <= r)
        return segmentTree[node];

    // Returns maximum number of divisors from left
    // or right subtree
    return max(query(2 * node, a, (a + b) / 2, l, r),
               query(2 * node + 1, ((a + b) / 2) + 1, b,l,r));
}

// driver code
int main()
{
    // First find smallest prime divisors for all
    // the numbers
    findSmallestPrimeFactors();

    // Then build the divisors[] array to store
    // the number of divisors
    buildDivisorsArray();

    // Build segment tree for the divisors[] array
    buildSegtmentTree(1, 1, maxn - 1);

    cout << "Maximum divisors that a number has "
         << " in [1, 100] are "
         << query(1, 1, maxn - 1, 1, 100) << endl;


    cout << "Maximum divisors that a number has"
         << " in [10, 48] are "
         << query(1, 1, maxn - 1, 10, 48) << endl;


    cout << "Maximum divisors that a number has"
         << " in [1, 10] are "
         << query(1, 1, maxn - 1, 1, 10) << endl;

    return 0;
}

Java

// Java implementation of the above idea to process
// queries of finding a number with maximum divisors.
import java.util.*;

class GFG
{
static int maxn = 10005;
static int INF = 999999;

static int []smallest_prime = new int[maxn];
static int []divisors = new int[maxn];
static int []segmentTree = new int[4 * maxn];

// Finds smallest prime factor of all numbers
// in range[1, maxn) and stores them in
// smallest_prime[], smallest_prime[i] should
// contain the smallest prime that divides i
static void findSmallestPrimeFactors()
{
    // Initialize the smallest_prime factors
    // of all to infinity
    for (int i = 0 ; i < maxn ; i ++ )
        smallest_prime[i] = INF;

    // to be built like eratosthenes sieve
    for (int i = 2; i < maxn; i++)
    {
        if (smallest_prime[i] == INF)
        {
            // prime number will have its
            // smallest_prime equal to itself
            smallest_prime[i] = i;
            for (int j = i * i; j < maxn; j += i)

                // if 'i' is the first
                // prime number reaching 'j'
                if (smallest_prime[j] > i)
                    smallest_prime[j] = i;
        }
    }
}

// number of divisors of n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
// are equal to (k1+1) * (k2+1) ... (kn+1)
// this function finds the number of divisors of all numbers
// in range [1, maxn) and stores it in divisors[]
// divisors[i] stores the number of divisors i has
static void buildDivisorsArray()
{
    for (int i = 1; i < maxn; i++)
    {
        divisors[i] = 1;
        int n = i, p = smallest_prime[i], k = 0;

        // we can obtain the prime factorization of
        // the number n, n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
        // using the smallest_prime[] array, we keep dividing n
        // by its smallest_prime until it becomes 1,
        // whilst we check if we have need to set k zero
        while (n > 1)
        {
            n = n / p;
            k ++;

            if (smallest_prime[n] != p)
            {
                // use p^k, initialize k to 0
                divisors[i] = divisors[i] * (k + 1);
                k = 0;
            }
            p = smallest_prime[n];
        }
    }
}

// builds segment tree for divisors[] array
static void buildSegtmentTree(int node, int a, int b)
{
    // leaf node
    if (a == b)
    {
        segmentTree[node] = divisors[a];
        return ;
    }

    //build left and right subtree
    buildSegtmentTree(2 * node, a, (a + b) / 2);
    buildSegtmentTree(2 * node + 1, ((a + b) / 2) + 1, b);

    //combine the information from left
    //and right subtree at current node
    segmentTree[node] = Math.max(segmentTree[2 * node],
                                 segmentTree[2 *node + 1]);
}

// returns the maximum number of divisors in [l, r]
static int query(int node, int a, int b, int l, int r)
{
    // If current node's range is disjoint
    // with query range
    if (l > b || a > r)
        return -1;

    // If the current node stores information
    // for the range that is completely inside
    // the query range
    if (a >= l && b <= r)
        return segmentTree[node];

    // Returns maximum number of divisors from left
    // or right subtree
    return Math.max(query(2 * node, a, (a + b) / 2, l, r),
                    query(2 * node + 1,
                         ((a + b) / 2) + 1, b, l, r));
}

// Driver Code
public static void main(String[] args)
{

    // First find smallest prime divisors
    // for all the numbers
    findSmallestPrimeFactors();

    // Then build the divisors[] array to store
    // the number of divisors
    buildDivisorsArray();

    // Build segment tree for the divisors[] array
    buildSegtmentTree(1, 1, maxn - 1);

    System.out.println("Maximum divisors that a number " +
                       "has in [1, 100] are " +
                        query(1, 1, maxn - 1, 1, 100));


    System.out.println("Maximum divisors that a number " +
                       "has in [10, 48] are " +
                        query(1, 1, maxn - 1, 10, 48));


    System.out.println("Maximum divisors that a number " +
                       "has in [1, 10] are " +
                        query(1, 1, maxn - 1, 1, 10));
    }
}

// This code is contributed by PrinciRaj1992

Python 3

# Python 3 implementation of the above
# idea to process queries of finding a
# number with maximum divisors.
maxn = 1000005
INF = 99999999

smallest_prime = [0] * maxn
divisors = [0] * maxn
segmentTree = [0] * (4 * maxn)

# Finds smallest prime factor of all
# numbers in range[1, maxn) and stores
# them in smallest_prime[], smallest_prime[i]
# should contain the smallest prime that divides i
def findSmallestPrimeFactors():

    # Initialize the smallest_prime
    # factors of all to infinity
    for i in range(maxn ):
        smallest_prime[i] = INF

    # to be built like eratosthenes sieve
    for i in range(2, maxn):
        if (smallest_prime[i] == INF):

            # prime number will have its
            # smallest_prime equal to itself
            smallest_prime[i] = i
            for j in range(i * i, maxn , i):

                # if 'i' is the first prime
                # number reaching 'j'
                if (smallest_prime[j] > i):
                    smallest_prime[j] = i

# number of divisors of n = (p1 ^ k1) *
# (p2 ^ k2) ... (pn ^ kn) are equal to
# (k1+1) * (k2+1) ... (kn+1). This function
# finds the number of divisors of all numbers
# in range [1, maxn) and stores it in divisors[]
# divisors[i] stores the number of divisors i has
def buildDivisorsArray():
    for i in range(1, maxn):
        divisors[i] = 1
        n = i
        p = smallest_prime[i]
        k = 0

        # we can obtain the prime factorization
        # of the number n n = (p1 ^ k1) * (p2 ^ k2)
        # ... (pn ^ kn) using the smallest_prime[]
        # array, we keep dividing n by its
        # smallest_prime until it becomes 1, whilst
        # we check if we have need to set k zero
        while (n > 1):
            n = n // p
            k += 1

            if (smallest_prime[n] != p):

                # use p^k, initialize k to 0
                divisors[i] = divisors[i] * (k + 1)
                k = 0

            p = smallest_prime[n]

# builds segment tree for divisors[] array
def buildSegtmentTree( node, a, b):

    # leaf node
    if (a == b):
        segmentTree[node] = divisors[a]
        return

    #build left and right subtree
    buildSegtmentTree(2 * node, a, (a + b) // 2)
    buildSegtmentTree(2 * node + 1,
                     ((a + b) // 2) + 1, b)

    #combine the information from left
    #and right subtree at current node
    segmentTree[node] = max(segmentTree[2 * node],
                            segmentTree[2 * node + 1])

# returns the maximum number of
# divisors in [l, r]
def query(node, a, b, l, r):

    # If current node's range is disjoint
    # with query range
    if (l > b or a > r):
        return -1

    # If the current node stores information
    # for the range that is completely inside
    # the query range
    if (a >= l and b <= r):
        return segmentTree[node]

    # Returns maximum number of divisors
    # from left or right subtree
    return max(query(2 * node, a, (a + b) // 2, l, r),
               query(2 * node + 1,
                    ((a + b) // 2) + 1, b, l, r))

# Driver code
if __name__ == "__main__":

    # First find smallest prime divisors
    # for all the numbers
    findSmallestPrimeFactors()

    # Then build the divisors[] array to
    # store the number of divisors
    buildDivisorsArray()

    # Build segment tree for the divisors[] array
    buildSegtmentTree(1, 1, maxn - 1)
    print("Maximum divisors that a number has ",
                            " in [1, 100] are ",
                  query(1, 1, maxn - 1, 1, 100))

    print("Maximum divisors that a number has",
                           " in [10, 48] are ",
                 query(1, 1, maxn - 1, 10, 48))


    print( "Maximum divisors that a number has",
                             " in [1, 10] are ",
                   query(1, 1, maxn - 1, 1, 10))

# This code is contributed by ita_c

C#

// C# implementation of the above idea
// to process queries of finding a number
// with maximum divisors.
using System;

class GFG
{
static int maxn = 10005;
static int INF = 999999;

static int []smallest_prime = new int[maxn];
static int []divisors = new int[maxn];
static int []segmentTree = new int[4 * maxn];

// Finds smallest prime factor of all numbers
// in range[1, maxn) and stores them in
// smallest_prime[], smallest_prime[i] should
// contain the smallest prime that divides i
static void findSmallestPrimeFactors()
{
    // Initialize the smallest_prime
    // factors of all to infinity
    for (int i = 0 ; i < maxn ; i ++ )
        smallest_prime[i] = INF;

    // to be built like eratosthenes sieve
    for (int i = 2; i < maxn; i++)
    {
        if (smallest_prime[i] == INF)
        {
            // prime number will have its
            // smallest_prime equal to itself
            smallest_prime[i] = i;
            for (int j = i * i; j < maxn; j += i)

                // if 'i' is the first
                // prime number reaching 'j'
                if (smallest_prime[j] > i)
                    smallest_prime[j] = i;
        }
    }
}

// number of divisors of
// n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
// are equal to (k1+1) * (k2+1) ... (kn+1)
// this function finds the number of divisors
// of all numbers in range [1, maxn) and stores
// it in divisors[] divisors[i] stores the
// number of divisors i has
static void buildDivisorsArray()
{
    for (int i = 1; i < maxn; i++)
    {
        divisors[i] = 1;
        int n = i, p = smallest_prime[i], k = 0;

        // we can obtain the prime factorization of
        // the number n,
        // n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
        // using the smallest_prime[] array,
        // we keep dividing n by its smallest_prime
        // until it becomes 1, whilst we check if
        // we have need to set k zero
        while (n > 1)
        {
            n = n / p;
            k ++;

            if (smallest_prime[n] != p)
            {
                // use p^k, initialize k to 0
                divisors[i] = divisors[i] * (k + 1);
                k = 0;
            }
            p = smallest_prime[n];
        }
    }
}

// builds segment tree for divisors[] array
static void buildSegtmentTree(int node,
                              int a, int b)
{
    // leaf node
    if (a == b)
    {
        segmentTree[node] = divisors[a];
        return;
    }

    //build left and right subtree
    buildSegtmentTree(2 * node, a, (a + b) / 2);
    buildSegtmentTree(2 * node + 1,
                    ((a + b) / 2) + 1, b);

    //combine the information from left
    //and right subtree at current node
    segmentTree[node] = Math.Max(segmentTree[2 * node],
                                 segmentTree[2 *node + 1]);
}

// returns the maximum number of divisors in [l, r]
static int query(int node, int a, int b, int l, int r)
{
    // If current node's range is disjoint
    // with query range
    if (l > b || a > r)
        return -1;

    // If the current node stores information
    // for the range that is completely inside
    // the query range
    if (a >= l && b <= r)
        return segmentTree[node];

    // Returns maximum number of divisors from left
    // or right subtree
    return Math.Max(query(2 * node, a, (a + b) / 2, l, r),
                    query(2 * node + 1,
                        ((a + b) / 2) + 1, b, l, r));
}

// Driver Code
public static void Main(String[] args)
{

    // First find smallest prime divisors
    // for all the numbers
    findSmallestPrimeFactors();

    // Then build the divisors[] array
    // to store the number of divisors
    buildDivisorsArray();

    // Build segment tree for the divisors[] array
    buildSegtmentTree(1, 1, maxn - 1);

    Console.WriteLine("Maximum divisors that a number " +
                                 "has in [1, 100] are " +
                          query(1, 1, maxn - 1, 1, 100));

    Console.WriteLine("Maximum divisors that a number " +
                                 "has in [10, 48] are " +
                          query(1, 1, maxn - 1, 10, 48));

    Console.WriteLine("Maximum divisors that a number " +
                                  "has in [1, 10] are " +
                           query(1, 1, maxn - 1, 1, 10));
    }
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// JavaScript implementation of the above idea to process
// queries of finding a number with maximum divisors.

    let maxn = 10005;
    let INF = 999999;
    let smallest_prime = new Array(maxn);
    for(let i=0;i<maxn;i++)
    {
        smallest_prime[i]=0;
    }
    let divisors = new Array(maxn);
    for(let i=0;i<maxn;i++)
    {
        divisors[i]=0;
    }
    let segmentTree = new Array(4 * maxn);
    for(let i=0;i<4*maxn;i++)
    {
        segmentTree[i]=0;
    }

    // Finds smallest prime factor of all numbers
    // in range[1, maxn) and stores them in
    // smallest_prime[], smallest_prime[i] should
    // contain the smallest prime that divides i
    function findSmallestPrimeFactors()
    {
        // Initialize the smallest_prime factors
        // of all to infinity
        for (let i = 0 ; i < maxn ; i ++ )
            smallest_prime[i] = INF;

        // to be built like eratosthenes sieve
        for (let i = 2; i < maxn; i++)
        {
            if (smallest_prime[i] == INF)
            {
                // prime number will have its
                // smallest_prime equal to itself
                smallest_prime[i] = i;
                for (let j = i * i; j < maxn; j += i)
                {
                    // if 'i' is the first
                    // prime number reaching 'j'
                    if (smallest_prime[j] > i)
                        smallest_prime[j] = i;
                }
            }
        }
    }

    // number of divisors of n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
    // are equal to (k1+1) * (k2+1) ... (kn+1)
    // this function finds the number of divisors of all numbers
    // in range [1, maxn) and stores it in divisors[]
    // divisors[i] stores the number of divisors i has
    function buildDivisorsArray()
    {
        for (let i = 1; i < maxn; i++)
        {
            divisors[i] = 1;
            let n = i;
            let p = smallest_prime[i]
            let k = 0;

            // we can obtain the prime factorization of
            // the number n, n = (p1 ^ k1) * (p2 ^ k2) ... (pn ^ kn)
            // using the smallest_prime[] array, we keep dividing n
            // by its smallest_prime until it becomes 1,
            // whilst we check if we have need to set k zero
            while (n > 1)
            {
                n = Math.floor(n / p);
                k++;

                if (smallest_prime[n] != p)
                {
                    // use p^k, initialize k to 0
                    divisors[i] = divisors[i] * (k + 1);
                    k = 0;
                }
                p = smallest_prime[n];
            }
        }
    }

    // builds segment tree for divisors[] array
    function buildSegtmentTree(node,a,b)
    {
        // leaf node
        if (a == b)
        {
            segmentTree[node] = divisors[a];
            return ;
        }

        //build left and right subtree
        buildSegtmentTree(2 * node, a, Math.floor((a + b) / 2));
        buildSegtmentTree((2 * node) + 1, Math.floor((a + b) / 2) + 1, b);

        //combine the information from left
        //and right subtree at current node
        segmentTree[node] = Math.max(segmentTree[2 * node],
                                     segmentTree[(2 *node) + 1]);
    }

    // returns the maximum number of divisors in [l, r]
    function query(node,a,b,l,r)
    {
        // If current node's range is disjoint
        // with query range
        if (l > b || a > r)
            return -1;

        // If the current node stores information
        // for the range that is completely inside
        // the query range
        if (a >= l && b <= r)
            return segmentTree[node];

        // Returns maximum number of divisors from left
        // or right subtree
        return Math.max(query(2 * node, a,
        Math.floor((a + b) / 2), l, r),
                        query(2 * node + 1,
                        Math.floor((a + b) / 2) + 1, b, l, r));
    }
    // Driver Code

    // First find smallest prime divisors
    // for all the numbers
    findSmallestPrimeFactors();

    // Then build the divisors[] array to store
    // the number of divisors
    buildDivisorsArray();

    // Build segment tree for the divisors[] array
    buildSegtmentTree(1, 1, maxn - 1);

    document.write("Maximum divisors that a number " +
    "has in [1, 100] are " + query(1, 1, maxn - 1, 1, 100)+"<br>");


    document.write("Maximum divisors that a number " +
                       "has in [10, 48] are " +
                        query(1, 1, maxn - 1, 10, 48)+"<br>");


    document.write("Maximum divisors that a number " +
                       "has in [1, 10] are " +
                        query(1, 1, maxn - 1, 1, 10)+"<br>");

    //  This code is contributed by avanitrachhadiya2155

</script>

Output:

Maximum divisors that a number has in [1, 100] are 12
Maximum divisors that a number has in [10, 48] are 10
Maximum divisors that a number has in [1, 10] are 4

Time Complexity: O((maxn + Q) * log(maxn))

• For sieve: O(maxn * log(log(maxn)) )
• For calculating divisors of each number: O(k_{1 + k2 + … + kn)} < O(log(maxn))
• For querying each range: O(log(maxn))

Auxiliary Space: O(n)

Related Topic: Segment Tree