math Namespace Reference

for std::vector More...

Functions
uint64_t	largestPower (uint32_t n, const uint16_t &p)
	Function to calculate largest power. More...
uint64_t	lcmSum (const uint16_t &num)
bool	magic_number (const uint64_t &n)
uint64_t	power (uint64_t a, uint64_t b, uint64_t c)
	This function calculates a raised to exponent b under modulo c using modular exponentiation. More...
template<class T >
T	n_choose_r (T n, T r)
	This is the function implementation of \( \binom{n}{r} \). More...
void	power_of_two (int n)
	Function to test above algorithm. More...
uint64_t	binomialCoeffSum (uint64_t n)

Detailed Description

for std::vector

Math algorithms.

for std::cin and std::cout

for std::cout

for IO operations

for io operations

for assert

for assert for std::cin and std::cout

Mathematical algorithms

for std::cin and std::cout

Mathematical algorithms

for assert

Mathematical algorithms

for assert for io operations

Mathematical algorithms

Mathematical algorithms

Function Documentation

binomialCoeffSum()

uint64_t math::binomialCoeffSum

(

uint64_t

n

)

Function to calculate sum of binomial coefficients

Parameters

n

number

Returns

Sum of binomial coefficients of number

                                      {
    // Calculating 2^n
    return (1 << n);
}

largestPower()

uint64_t math::largestPower	(	uint32_t	n,
		const uint16_t &	p
	)

Function to calculate largest power.

Parameters

n	number
p	prime number

Returns

largest power

    {  
        // Initialize result  
        int x = 0;  
  
        // Calculate result 
        while (n)  
        {  
            n /= p;  
            x += n;  
        }  
        return x;  
    }

lcmSum()

uint64_t math::lcmSum

(

const uint16_t &

num

)

Function to compute sum of euler totients in sumOfEulerTotient vector

Parameters

num	input number

Returns

int Sum of LCMs, i.e. ∑LCM(i, num) from i = 1 to num

                                     {
    uint64_t i = 0, j = 0;
    std::vector<uint64_t> eulerTotient(num + 1);
    std::vector<uint64_t> sumOfEulerTotient(num + 1);
 
    // storing initial values in eulerTotient vector
    for (i = 1; i <= num; i++) {
        eulerTotient[i] = i;
    }
 
    // applying totient sieve
    for (i = 2; i <= num; i++) {
        if (eulerTotient[i] == i) {
            for (j = i; j <= num; j += i) {
                eulerTotient[j] = eulerTotient[j] / i;
                eulerTotient[j] = eulerTotient[j] * (i - 1);
            }
        }
    }
 
    // computing sum of euler totients
    for (i = 1; i <= num; i++) {
        for (j = i; j <= num; j += i) {
            sumOfEulerTotient[j] += eulerTotient[i] * i;
        }
    }
 
    return ((sumOfEulerTotient[num] + 1) * num) / 2;
}

magic_number()

bool math::magic_number

(

const uint64_t &

n

)

Function to check if the given number is magic number or not.

Parameters

n	number to be checked.

Returns

if number is a magic number, returns true, else false.

                                     {
    if (n <= 0) {
        return false;
    }
    // result stores the modulus of @param n with 9
    uint64_t result = n % 9;
    // if result is 1 then the number is a magic number else not
    if (result == 1) {
        return true;
    } else {
        return false;
    }
}

n_choose_r()

template<class T >

T math::n_choose_r	(	T	n,
		T	r
	)

This is the function implementation of \( \binom{n}{r} \).

We are calculating the ans with iterations instead of calculating three different factorials. Also, we are using the fact that \( \frac{n!}{r! (n-r)!} = \frac{(n - r + 1) \times \cdots \times n}{1 \times \cdots \times r} \)

Template Parameters

T	Only for integer types such as long, int_64 etc

Parameters

n	\( n \) in \( \binom{n}{r} \)
r	\( r \) in \( \binom{n}{r} \)

Returns

ans \( \binom{n}{r} \)

                       {
    if (r > n / 2) {
        r = n - r;  // Because of the fact that  nCr(n, r) == nCr(n, n - r)
    }
    T ans = 1;
    for (int i = 1; i <= r; i++) {
        ans *= n - r + i;
        ans /= i;
    }
    return ans;
}

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power()

uint64_t math::power	(	uint64_t	a,
		uint64_t	b,
		uint64_t	c
	)

This function calculates a raised to exponent b under modulo c using modular exponentiation.

Parameters

a	integer base
b	unsigned integer exponent
c	integer modulo

Returns

a raised to power b modulo c

Initialize the answer to be returned

Update a if it is more than or equal to c

In case a is divisible by c;

If b is odd, multiply a with answer

b must be even now

b = b/2

                                                   {
    uint64_t ans = 1;  /// Initialize the answer to be returned
    a = a % c;         /// Update a if it is more than or equal to c
    if (a == 0) {
        return 0;  /// In case a is divisible by c;
    }
    while (b > 0) {
        /// If b is odd, multiply a with answer
        if (b & 1) {
            ans = ((ans % c) * (a % c)) % c;
        }
        /// b must be even now
        b = b >> 1;  /// b = b/2
        a = ((a % c) * (a % c)) % c;
    }
    return ans;
}

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power_of_two()

void math::power_of_two

(

int

n

)

Function to test above algorithm.

Parameters

n	description

Returns

void

This function finds whether a number is power of 2 or not

Parameters

n	value for which we want to check prints the result, as "Yes, the number n is a power of 2" or "No, the number is not a power of 2" without quotes

result stores the bitwise and of n and n-1

                         {
    /**
     * This function finds whether a number is power of 2 or not
     * @param n value for which we want to check
     * prints the result, as "Yes, the number n is a power of 2" or
     * "No, the number is not a power of 2" without quotes
     */
    /// result stores the
    /// bitwise and of n and n-1
    int result = n & (n - 1);
    if (result == 0) {
        std::cout << "Yes, the number " << n << " is a power of 2";
    } else {
        std::cout << "No, the number " << n << " is not a power of 2";
    }
}