Step by Step solution about How to resolve the algorithm Abundant odd numbers step by step in the C programming language

The provided C code finds and prints abundant odd numbers. An abundant number is a positive integer for which the sum of its proper divisors (the positive divisors of the number, excluding the number itself) is greater than the number itself.

Here's a breakdown of the code:

1. Function sum_proper_divisors:

   • Input: Takes an unsigned integer n.
   • Purpose: Calculates the sum of proper divisors of n for odd numbers only. It uses a loop with an increment of 2 to consider only odd numbers.
   • Implementation:
     • Initializes sum to 1, which accounts for the divisor 1.
     • Iterates from 3 to the square root of n with an increment of 2.
     • Checks if n is divisible by i (using n % i == 0).
     • If divisible, adds i to sum.
     • If i is equal to j, which is the quotient of n divided by i, it means that i and j are not different divisors (e.g., 4 and 2), so only one of them is added to the sum.
   • Returns the calculated sum of proper divisors.

2. Main Function:

   • Input: Accepts two command-line arguments (unused in this code).
   • Purpose: Finds and prints abundant odd numbers.
   • Implementation:
     • Initializes n to 1 and c to 0.
     • Enters a loop that continues until c reaches 25.
     • Inside the loop:
       • Increments n by 2 to consider only odd numbers.
       • Checks if n is less than the sum of its proper divisors (using n < sum_proper_divisors(n)).
       • If n is abundant, prints n and increments c.
     • After printing the first 25 abundant odd numbers, the loop continues without printing to find the 1000th abundant odd number.
     • Finally, the loop breaks when the first abundant odd number above one billion is found.

3. Output:

   • The program prints the first 25 abundant odd numbers.
   • It then prints the 1000th abundant odd number.
   • Finally, it prints the first abundant odd number above one billion.

#include <stdio.h>
#include <math.h>

// The following function is for odd numbers ONLY
// Please use "for (unsigned i = 2, j; i*i <= n; i ++)" for even and odd numbers
unsigned sum_proper_divisors(const unsigned n) {
  unsigned sum = 1;
  for (unsigned i = 3, j; i < sqrt(n)+1; i += 2) if (n % i == 0) sum += i + (i == (j = n / i) ? 0 : j);
  return sum;
}

int main(int argc, char const *argv[]) {
  unsigned n, c;
  for (n = 1, c = 0; c < 25; n += 2) if (n < sum_proper_divisors(n)) printf("%u: %u\n", ++c, n);

  for ( ; c < 1000; n += 2) if (n < sum_proper_divisors(n)) c ++;
  printf("\nThe one thousandth abundant odd number is: %u\n", n);

  for (n = 1000000001 ;; n += 2) if (n < sum_proper_divisors(n)) break;
  printf("The first abundant odd number above one billion is: %u\n", n);
  
  return 0;
}