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# Expressing factorial n as sum of consecutive numbers

• Difficulty Level : Hard
• Last Updated : 10 Jan, 2023

Given two numbers N and M. Find the number of ways in which factorial N can be expressed as a sum of two or more consecutive numbers. Print the result modulo M.
Examples:

Input : N = 3, M = 7
Output : 1
Explanation:  3! can be expressed
in one way, i.e. 1 + 2 + 3 = 6.
Hence 1 % 7 = 1

Input : N = 4, M = 7
Output : 1
Explanation:  4! can be expressed
in one way, i.e. 7 + 8 + 9 = 24
Hence 1 % 7 = 1

A simple solution is to first compute factorial, then count number of ways to represent factorial as sum of consecutive numbers using Count ways to express a number as sum of consecutive numbers. This solution causes overflow.
Below is a better solution to avoid overflow.
Let us consider that sum of r consecutive numbers be expressed as:
(a + 1) + (a + 2) + (a + 3) + … + (a + r), which simplifies as (r * (r + 2*a + 1)) / 2
Hence, (a + 1) + (a + 2) + (a + 3) + … + (a + r) = (r * (r + 2*a + 1)) / 2. Since the above expression is equal to factorial N, we write it as
2 * N! = r * (r + 2*a + 1)
Instead of counting all the pairs (r, a), we will count all pairs (r, r + 2*a + 1). Now, we are just counting all ordered pairs (X, Y) with XY = 2 * N! where X < Y and X, Y have different parity, that means if (r) is even, (r + 2*a + 1) is odd or if (r) is odd then (r + 2*a + 1) is even. This is equivalent to finding the odd divisors of 2 * N! which will be same as odd divisors of N!.
For counting the number of divisors in N!, we calculate the power of primes in factorization and total count of divisors become (p1 + 1) * (p2 + 1) * … * (pn + 1). To calculate the largest power of a prime in N!, we will use legendre’s formula Below is the implementation of the above approach.

## C++

 // CPP program to count number of// ways we can express a factorial// as sum of consecutive numbers#include using namespace std; #define MAX 50002 vector<int> primes; // sieve of Eratosthenes to compute// the prime numbersvoid sieve(){    bool isPrime[MAX];    memset(isPrime, true, sizeof(isPrime));     for (int p = 2; p * p < MAX; p++) {        if (isPrime[p] == true) {            for (int i = p * 2; i < MAX; i += p)                isPrime[i] = false;        }    }     // Store all prime numbers    for (int p = 2; p < MAX; p++)        if (isPrime[p])            primes.push_back(p);} // function to calculate the largest// power of a prime in a numberlong long int power(long long int x,                    long long int y){    long long int count = 0;    long long int z = y;    while (x >= z) {        count += (x / z);        z *= y;    }    return count;} // Modular multiplication to avoid// the overflow of multiplication// Please see below for details// https://www.geeksforgeeks.org/how-to-avoid-overflow-in-modular-multiplication/long long int modMult(long long int a,                      long long int b,                      long long int mod){    long long int res = 0;    a = a % mod;    while (b > 0) {        if (b % 2 == 1)            res = (res + a) % mod;        a = (a * 2) % mod;        b /= 2;    }    return res % mod;} // Returns count of ways to express n!// as sum of consecutives.long long int countWays(long long int n,                        long long int m){    long long int ans = 1;     // We skip 2 (First prime) as we need to    // consider only odd primes    for (int i = 1; i < primes.size(); i++) {         // compute the largest power of prime        long long int powers = power(n, primes[i]);         // if the power of current prime number        // is zero in N!, power of primes greater        // than current prime number will also        // be zero, so break out from the loop        if (powers == 0)            break;         // multiply the result at every step        ans = modMult(ans, powers + 1, m) % m;    }     // subtract 1 to exclude the case of 1    // being an odd divisor    if (((ans - 1) % m) < 0)        return (ans - 1 + m) % m;    else        return (ans - 1) % m;} // Driver Codeint main(){    sieve();    long long int n = 4, m = 7;    cout << countWays(n, m);    return 0;}

## Java

 // Java program to count number of// ways we can express a factorial// as sum of consecutive numbersimport java.util.*; class GFG {         static int MAX = 50002;    static ArrayList primes                 = new ArrayList();                      // sieve of Eratosthenes to compute    // the prime numbers    public static void sieve()    {                 boolean isPrime[] = new boolean[MAX];                 for(int i = 0; i < MAX; i++)            isPrime[i] = true;                     for (int p = 2; p * p < MAX; p++) {            if (isPrime[p] == true) {                for (int i = p * 2; i < MAX; i += p)                    isPrime[i] = false;            }        }                 // Store all prime numbers        for (int p = 2; p < MAX; p++)            if (isPrime[p] == true)                primes.add(p);    }             // function to calculate the largest    // power of a prime in a number    public static int power(int x, int y)    {        int count = 0;        int z = y;        while (x >= z) {            count += (x / z);            z *= y;        }                 return count;    }         // Modular multiplication to avoid    // the overflow of multiplication    // Please see below for details    // https://www.geeksforgeeks.org/how-to-avoid-overflow-in-modular-multiplication/    public static int modMult(int a, int b, int mod)    {        int res = 0;        a = a % mod;        while (b > 0) {            if (b % 2 == 1)                res = (res + a) % mod;            a = (a * 2) % mod;            b /= 2;        }                 return res % mod;    }         // Returns count of ways to express n!    // as sum of consecutives.    public static int countWays(int n, int m)    {        int ans = 1;             // We skip 2 (First prime) as we need to        // consider only odd primes        for (int i = 1; i < primes.size(); i++) {                 // compute the largest power of prime            int powers = power(n, primes.get(i));                 // if the power of current prime number            // is zero in N!, power of primes greater            // than current prime number will also            // be zero, so break out from the loop            if (powers == 0)                break;                 // multiply the result at every step            ans = modMult(ans, powers + 1, m) % m;        }             // subtract 1 to exclude the case of 1        // being an odd divisor        if (((ans - 1) % m) < 0)            return (ans - 1 + m) % m;        else            return (ans - 1) % m;    }         //Driver function    public static void main (String[] args) {                 sieve();                 int n = 4, m = 7;                 System.out.println(countWays(n,m));    }} // This code is contributed by akash1295.

## Python 3

 # Python 3 program to count number of# ways we can express a factorial# as sum of consecutive numbersMAX = 50002; primes = [] # sieve of Eratosthenes to compute# the prime numbersdef sieve():    isPrime = [True]*(MAX)         p = 2    while p * p < MAX :        if (isPrime[p] == True):            for i in range( p * 2,MAX, p):                isPrime[i] = False        p+=1     # Store all prime numbers    for p in range( 2,MAX):        if (isPrime[p]):            primes.append(p) # function to calculate the largest# power of a prime in a numberdef power( x, y):     count = 0    z = y    while (x >= z):        count += (x // z)        z *= y         return count # Modular multiplication to avoid# the overflow of multiplication# Please see below for details# https://www.geeksforgeeks.org/how-to-avoid-overflow-in-modular-multiplication/def modMult(a, b,mod):    res = 0    a = a % mod    while (b > 0):        if (b % 2 == 1):            res = (res + a) % mod        a = (a * 2) % mod        b //= 2         return res % mod # Returns count of ways to express n!# as sum of consecutives.def countWays(n,m):    ans = 1     # We skip 2 (First prime) as we need to    # consider only odd primes    for i in range(1,len(primes)):         # compute the largest power of prime        powers = power(n, primes[i])         # if the power of current prime number        # is zero in N!, power of primes greater        # than current prime number will also        # be zero, so break out from the loop        if (powers == 0):            break         # multiply the result at every step        ans = modMult(ans, powers + 1, m) % m          # subtract 1 to exclude the case of 1    # being an odd divisor    if (((ans - 1) % m) < 0):        return (ans - 1 + m) % m    else:        return (ans - 1) % m # Driver Codeif __name__ == "__main__":    sieve()    n = 4    m = 7    print(countWays(n, m))     # This code is contributed by ChitraNayal

## C#

 // C# program to count number of// ways we can express a factorial// as sum of consecutive numbersusing System ;using System.Collections; class GFG {         static int MAX = 50002;    static ArrayList primes = new ArrayList ();                     // sieve of Eratosthenes to compute    // the prime numbers    public static void sieve()    {                 bool []isPrime = new bool[MAX];                 for(int i = 0; i < MAX; i++)            isPrime[i] = true;                     for (int p = 2; p * p < MAX; p++) {            if (isPrime[p] == true) {                for (int i = p * 2; i < MAX; i += p)                    isPrime[i] = false;            }        }                 // Store all prime numbers        for (int p = 2; p < MAX; p++)            if (isPrime[p] == true)                primes.Add(p);    }             // function to calculate the largest    // power of a prime in a number    public static int power_prime(int x, int y)    {        int count = 0;        int z = y;        while (x >= z) {            count += (x / z);            z *= y;        }                 return count;    }         // Modular multiplication to avoid    // the overflow of multiplication    // Please see below for details    // https://www.geeksforgeeks.org/how-to-avoid-overflow-in-modular-multiplication/    public static int modMult(int a, int b, int mod)    {        int res = 0;        a = a % mod;        while (b > 0) {            if (b % 2 == 1)                res = (res + a) % mod;            a = (a * 2) % mod;            b /= 2;        }                 return res % mod;    }         // Returns count of ways to express n!    // as sum of consecutives.    public static int countWays(int n, int m)    {        int ans = 1;             // We skip 2 (First prime) as we need to        // consider only odd primes        for (int i = 1; i < primes.Count; i++) {                 // compute the largest power of prime            int powers = power_prime(n, Convert.ToInt32(primes[i]));                 // if the power of current prime number            // is zero in N!, power of primes greater            // than current prime number will also            // be zero, so break out from the loop            if (powers == 0)                break;                 // multiply the result at every step            ans = modMult(ans, powers + 1, m) % m;        }             // subtract 1 to exclude the case of 1        // being an odd divisor        if (((ans - 1) % m) < 0)            return (ans - 1 + m) % m;        else            return (ans - 1) % m;    }         //Driver function    public static void Main () {                 sieve();                 int n = 4, m = 7;                 Console.WriteLine(countWays(n,m));    }} // This code is contributed by Ryuga

## Javascript

 

Output:

1

Time Complexity: O(MAX*log(log(MAX))+M*log(K)) where MAX=5002, M is the number of primes less than MAX(i.e 5002), and K is the greatest prime number less than MAX.

Auxiliary Space: O(MAX)

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