Prime factorization of a factorial

Question

I need to write a program to input a number and output its factorial\'s prime factorization in the form:

4!=(2^3)*(3^1)

5!=(2^3)*(3^1)*(5^1)

Answer 1

You can use O(n/2 log log n) algorithm using only sums (no need precalc primes).

This is a sieve using relation

f = a * b  ~>  f^k = a^k * b^k

then, we reduce all initial factors 1 * 2 * 3 * ... * n moving k from big numbers to small numbers.

Using Sieve of Atkin the Will Ness algorithm could be better for very big n if not, I think it could be better

#include 
#include 

int main(int argc, char **argv) {
  int n = atoi(argv[1]);
  int *p = (int *) malloc(sizeof(int) * (n + 1));
  int i, j, d;
  for(i = 0; i <= n; i++)
    p[i] = 1;
  for(i = n >> 1; i > 1; i--)
    if(p[i]) {
      for(j = i + i, d = 2; j <= n; j += i, d++) {
        if(p[j]) {
          p[i] += p[j];
          p[d] += p[j];
          p[j] = 0;
        }
      }
    }
  printf("1");
  for(i = 2; i <= n; i++)
    if(p[i])
      printf(" * %i^%i", i, p[i]);
  printf("\n");
  return 0;
}