Step by Step solution about How to resolve the algorithm Chowla numbers step by step in the C++ programming language

This C++ code is designed to explore the fascinating topics of Chowla numbers, prime numbers, and perfect numbers. Here's a detailed breakdown:

Chowla Numbers:

• The chowla function calculates the sum of the proper divisors of a given number n. Proper divisors are positive divisors of n that are less than n itself.
• The function iterates through numbers from 2 to the square root of n, checking if n is divisible by them.
• If n is divisible by a number i, the sum is incremented by both i and n/i (if they are different).

Sieve of Eratosthenes:

• The sieve function uses the Sieve of Eratosthenes algorithm to efficiently find prime numbers.
• It initializes a boolean vector c of size limit to false (assuming all numbers initially prime).
• Starting from 3, it marks multiples of each prime as composite (true).
• The sieve ensures that only odd numbers are checked for primality (even numbers are composite except 2).

Prime Counting:

• The main function counts prime numbers within certain limits.
• It uses the sieve function to find primes up to 10 million and prints the count at various power-of-10 intervals.

Perfect Numbers:

• A number is perfect if the sum of its proper divisors (excluding itself) equals the number itself.
• The code iterates through pairs of consecutive odd numbers (called "Euler pairs") and checks if the product of the pair is a perfect number.
• It uses the chowla function to efficiently calculate the sum of proper divisors and count the number of perfect numbers up to a limit of 35 million.

In summary, this code explores the mathematical concepts of Chowla numbers, prime numbers, and perfect numbers, providing insights into their properties and generating counts within specified limits. It uses efficient algorithms like the Sieve of Eratosthenes for prime counting and the Chowla function for finding the sum of proper divisors.

#include <vector>
#include <iostream>

using namespace std;

int chowla(int n)
{
	int sum = 0;
	for (int i = 2, j; i * i <= n; i++)
		if (n % i == 0) sum += i + (i == (j = n / i) ? 0 : j);
	return sum;
}

vector<bool> sieve(int limit)
{
	// True denotes composite, false denotes prime.
	// Only interested in odd numbers >= 3
	vector<bool> c(limit);
	for (int i = 3; i * 3 < limit; i += 2)
		if (!c[i] && (chowla(i) == 0))
			for (int j = 3 * i; j < limit; j += 2 * i)
				c[j] = true;
	return c;
}

int main()
{
	cout.imbue(locale(""));
	for (int i = 1; i <= 37; i++)
		cout << "chowla(" << i << ") = " << chowla(i) << "\n";
	int count = 1, limit = (int)(1e7), power = 100;
	vector<bool> c = sieve(limit);
	for (int i = 3; i < limit; i += 2)
	{
		if (!c[i]) count++;
		if (i == power - 1)
		{
			cout << "Count of primes up to " << power << " = "<< count <<"\n";
			power *= 10;
		}
	}

	count = 0; limit = 35000000;
	int k = 2, kk = 3, p;
	for (int i = 2; ; i++)
	{
		if ((p = k * kk) > limit) break;
		if (chowla(p) == p - 1)
		{
			cout << p << " is a number that is perfect\n";
			count++;
		}
		k = kk + 1; kk += k;
	}
	cout << "There are " << count << " perfect numbers <= 35,000,000\n";
	return 0;
}