Step by Step solution about How to resolve the algorithm Yellowstone sequence step by step in the C++ programming language

The provided C++ code implements a generator for the Yellowstone numbers, which are a sequence of positive integers defined by the following set of rules:

1. 1 and 2 are Yellowstone numbers.
2. For any integer n >= 3, n is a Yellowstone number if and only if:
   • n does not belong to the set of already generated Yellowstone numbers.
   • The greatest common divisor (GCD) of n with the previous Yellowstone number (denoted by n1) is 1.
   • The GCD of n with the second-to-last Yellowstone number (denoted by n2) is greater than 1.

The yellowstone_generator class is a template class that generates Yellowstone numbers of a given integer type. It maintains a set of generated Yellowstone numbers (sequence_) and three private member variables (min_, n_, and n1_, n2_) to keep track of the current and previous Yellowstone numbers.

The next() method generates the next Yellowstone number. It first updates n2_ to n1_ and n1_ to n_. Then, it checks if n_ is less than 3. If n_ is less than 3, it simply increments n_ and returns it.

If n_ is greater than or equal to 3, it enters a loop to find the next Yellowstone number. It starts by incrementing n_ until it finds a number that satisfies the three conditions mentioned earlier. Once it finds a suitable number, it inserts it into the sequence_ set and removes all the numbers less than min_ from the set. It then increments min_ and returns the newly generated Yellowstone number.

The main() function creates an instance of the yellowstone_generator class with an unsigned int type and prints the first 30 Yellowstone numbers.

Here is a step-by-step explanation of the code:

1. The yellowstone_generator class is a template class that takes an integer type as a parameter.

2. The next() method of the yellowstone_generator class generates the next Yellowstone number. It does this by maintaining a set of generated Yellowstone numbers (sequence_), and three private member variables (min_, n_, and n1_, n2_) to keep track of the current, previous, and second-to-last Yellowstone numbers.

3. The next() method first updates n2_ to n1_ and n1_ to n_. Then, it checks if n_ is less than 3. If n_ is less than 3, it simply increments n_ and returns it.

4. If n_ is greater than or equal to 3, it enters a loop to find the next Yellowstone number. It starts by incrementing n_ until it finds a number that satisfies the three conditions mentioned earlier. Once it finds a suitable number, it inserts it into the sequence_ set and removes all the numbers less than min_ from the set. It then increments min_ and returns the newly generated Yellowstone number.

5. The main() function creates an instance of the yellowstone_generator class with an unsigned int type and prints the first 30 Yellowstone numbers.

Here is an example output of the program:

First 30 Yellowstone numbers:
1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109

#include <iostream>
#include <numeric>
#include <set>

template <typename integer>
class yellowstone_generator {
public:
    integer next() {
        n2_ = n1_;
        n1_ = n_;
        if (n_ < 3) {
            ++n_;
        } else {
            for (n_ = min_; !(sequence_.count(n_) == 0
                && std::gcd(n1_, n_) == 1
                && std::gcd(n2_, n_) > 1); ++n_) {}
        }
        sequence_.insert(n_);
        for (;;) {
            auto it = sequence_.find(min_);
            if (it == sequence_.end())
                break;
            sequence_.erase(it);
            ++min_;
        }
        return n_;
    }
private:
    std::set<integer> sequence_;
    integer min_ = 1;
    integer n_ = 0;
    integer n1_ = 0;
    integer n2_ = 0;
};

int main() {
    std::cout << "First 30 Yellowstone numbers:\n";
    yellowstone_generator<unsigned int> ygen;
    std::cout << ygen.next();
    for (int i = 1; i < 30; ++i)
        std::cout << ' ' << ygen.next();
    std::cout << '\n';
    return 0;
}