Calculating 1^X + 2^X + ... + N^X mod 1000000007

Question

Is there any algorithm to calculate (1^x + 2^x + 3^x + ... + n^x) mod 1000000007?
Note: a^b is the b-th power of a.

The constraints are 1 <= n <= 10^16, 1 <= x <= 1000. So the value of N is very large.

I can only solve for O(m log m) if m = 1000000007. It is very slow because the time limit is 2 secs.

Do you have any efficient algorithm?

There was a comment that it might be duplicate of this question, but it is definitely different.

Answer 1

You can sum up the series

1**X + 2**X + ... + N**X

with the help of Faulhaber's formula and you'll get a polynomial with an X + 1 power to compute for arbitrary N.

If you don't want to compute Bernoulli Numbers, you can find the the polynomial by solving X + 2 linear equations (for N = 1, N = 2, N = 3, ..., N = X + 2) which is a slower method but easier to implement.

Let's have an example for X = 2. In this case we have an X + 1 = 3 order polynomial:

    A*N**3 + B*N**2 + C*N + D

The linear equations are

      A +    B +   C + D = 1              =  1 
    A*8 +  B*4 + C*2 + D = 1 + 4          =  5
   A*27 +  B*9 + C*3 + D = 1 + 4 + 9      = 14
   A*64 + B*16 + C*4 + D = 1 + 4 + 9 + 16 = 30 

Having solved the equations we'll get

  A = 1/3
  B = 1/2
  C = 1/6
  D =   0 

The final formula is

  1**2 + 2**2 + ... + N**2 == N**3 / 3 + N**2 / 2 + N / 6

Now, all you have to do is to put an arbitrary large N into the formula. So far the algorithm has O(X**2) complexity (since it doesn't depend on N).