Step by Step solution about How to resolve the algorithm 9 billion names of God the integer step by step in the C programming language

This C code uses the GNU Multiple Precision Arithmetic Library (GMP) to calculate and print the values of a sequence of numbers defined as follows:

p[n] = 0
for k = 1 to n
 d = n - k * (3 * k - 1) / 2
 if d >= 0
   if k is odd
     p[n] += p[d]
   else
     p[n] -= p[d]

   d -= k
   if d >= 0
     if k is odd
       p[n] += p[d]
     else
       p[n] -= p[d]

Here's a detailed breakdown of the code:

1. Header Inclusions:

   • The code includes the necessary header files <stdio.h> for input and output and <gmp.h> for the GMP library.

2. Constant Definition:

   • A constant N is defined as 100,000, which represents the maximum index up to which the sequence will be calculated.

3. Global Array:

   • A global array p is declared to store the values of the sequence up to index N.

4. calc Function:

   • This function takes an integer n as input and calculates the value of p[n] based on the definition given above.
   • It uses GMP's mpz_init_set_ui to initialize p[n] to 0 and then iterates from k = 1 to k <= n.
   • Inside the loop, it calculates d and checks if d is non-negative. If d is valid, it performs the operations defined in the sequence.
   • It uses GMP's mpz_add and mpz_sub functions to add or subtract values from p[n] based on whether k is odd or even.

5. main Function:

   • The main function is the entry point of the program.
   • It initializes idx as an array of integers representing specific values of n for which the sequence will be printed.
   • It initializes at as 0, which keeps track of the index in the idx array.
   • It initializes p[0] to 1 using mpz_init_set_ui.

6. Calculation and Printing:

   • The code enters a loop that runs from i = 1 to the last value in idx.
   • For each i, it calls the calc function to calculate p[i].
   • If i is not equal to idx[at], it moves to the next iteration.
   • Otherwise, it prints i and the value of p[i] using gmp_printf.
   • It then increments at to move to the next value in the idx array.

This code essentially computes a sequence of values defined by a recursive formula and prints specific terms of the sequence based on the values provided in the idx array.

#include <stdio.h>
#include <gmp.h>

#define N 100000
mpz_t p[N + 1];

void calc(int n)
{
	mpz_init_set_ui(p[n], 0);

	for (int k = 1; k <= n; k++) {
		int d = n - k * (3 * k - 1) / 2;
		if (d < 0) break;

		if (k&1)mpz_add(p[n], p[n], p[d]);
		else	mpz_sub(p[n], p[n], p[d]);

		d -= k;
		if (d < 0) break;

		if (k&1)mpz_add(p[n], p[n], p[d]);
		else	mpz_sub(p[n], p[n], p[d]);
	}
}

int main(void)
{
	int idx[] = { 23, 123, 1234, 12345, 20000, 30000, 40000, 50000, N, 0 };
	int at = 0;

	mpz_init_set_ui(p[0], 1);

	for (int i = 1; idx[at]; i++) {
		calc(i);
		if (i != idx[at]) continue;

		gmp_printf("%2d:\t%Zd\n", i, p[i]);
		at++;
	}
}