Step by Step solution about How to resolve the algorithm Numbers which are the cube roots of the product of their proper divisors step by step in the C# programming language

The provided C# code determines whether a given positive integer n has exactly four or eight distinct positive divisors, including 1 and n itself. If n satisfies this condition, it then checks if n is the cube root of the product of all its proper divisors (divisors excluding n). The program continues this process until it finds the first 50 such numbers and prints them. Here's a detailed explanation of the code:

• The dc8 function takes an unsigned 32-bit integer n as an argument. It calculates the number of distinct positive divisors of n by grouping together consecutive powers of 2, 3, and other prime factors. It returns true if n has exactly four or eight distinct positive divisors, and false otherwise.
• The Main method initializes variables for counting, limiting, and printing. It then enters a loop that iterates through positive integers n starting from 1.
• For each n, it checks if n is equal to 1 or if it has exactly four or eight distinct positive divisors by calling the dc8 function. If either condition is met, it increments the count and prints n in a formatted way.
• The loop continues until it finds the first 50 numbers that meet the specified conditions, or reaches the limit lmt. Every time the count reaches lmt, the limit is multiplied by 10 to increase the search range.
• The program prints the first 50 numbers that satisfy the given conditions in a tabular format, with 10 numbers per line.

In summary, the code efficiently checks if a given number n has exactly four or eight distinct positive divisors and then determines if n is the cube root of the product of its proper divisors. It outputs the first 50 such numbers, providing insights into the properties of positive integers and their divisors.

using System;
class Program {

  static bool dc8(uint n) {
    uint res = 1, count, p, d;
    for ( ; (n & 1) == 0; n >>= 1) res++;
    for (count = 1; n % 3 == 0; n /= 3) count++;
    for (p = 5, d = 4; p * p <= n; p += d = 6 - d)
        for (res *= count, count = 1; n % p == 0; n /= p) count++;
    return n > 1 ? res * count == 4 : res * count == 8;
  }

  static void Main(string[] args) {
    Console.WriteLine("First 50 numbers which are the cube roots of the products of "
                      + "their proper divisors:");
    for (uint n = 1, count = 0, lmt = 500; count < 5e6; ++n) if (n == 1 || dc8(n))
        if (++count <= 50) Console.Write("{0,3}{1}",n, count % 10 == 0 ? '\n' : ' ');
        else if (count == lmt) Console.Write("{0,16:n0}th: {1:n0}\n", count, n, lmt *= 10);
  }
}