List of powers of Natural Numbers Greatings,
  Some time ago a friend of mine showed me this astonishing algorithm and recently i tried to find some information about it on the internet but couldn't find anything... Please help.
Pseudocode: 
Consider that 1 is the starting index of a list
1.  input natural number n.
2.  let s = list of all natural numbers {1, 2, 3, 4, 5, 6 ...}
3.  while (n>1) do
    3.1.  drop each n-th elementh from s 
    3.2.  for int i = 2 to ∞ do s[i] += s[i-1] 
    3.3.  n = n-1 
4.  Now s = {1n, 2n, 3n, 4n ...}

Example:
n = 3
s = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}
perform 3.1:  s = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16...} 
perform 3.2:  s = {1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91 ...} 
perform 3.3:  n = 2 > 1 
perform 3.1:  s = {1, 7, 19, 37, 61, 91 ...} 
perform 3.2:  s = {1, 8, 27, 64, 125, 216 ...} 
perform 3.3:  n = 1 => end of while loop 
The final state of s is {13, 23, 33, 43, 53 ...}

You can calculate the closed form formulas for sums of natural powers using finite differences. What I mean is this:
http://mathforum.org/library/drmath/view/56982.html
What this appears to be doing is directly calculating the the powers themselves using finite differences. 
