Problem:
Score : 400 points

Problem Statement

For a positive integer X, let f(X) be the number of positive divisors of X.

Given a positive integer N, find \sum_{K=1}^N K\times f(K).

Constraints

1 \leq N \leq 10^7

Input

Input is given from Standard Input in the following format:

N

Output

Print the value \sum_{K=1}^N K\times f(K).

Sample Input 1

4

Sample Output 1

23

We have f(1)=1, f(2)=2, f(3)=2, and f(4)=3, so the answer is 1\times 1 + 2\times 2 + 3\times 2 + 4\times 3 =23.

Sample Input 2

100

Sample Output 2

26879

Sample Input 3

10000000

Sample Output 3

838627288460105

Watch out for overflows.

Solution in OCaml (4.10.0) (judge verdict: Accepted)

let () = Scanf.scanf "%d" @@ fun n ->
  let sum = ref 1 in
  let dp = Array.make (n + 1) 1 in
  let is_prime = Array.make (n + 1) true in
  for p = 2 to n do
    if is_prime.(p) then begin
      let rec outer i pp =
        if pp <= n then begin
          let rec inner j =
            if j <= n then begin
              is_prime.(j) <- false;
              dp.(j) <- dp.(j) * (1 + i) / i;
              inner (j + pp)
            end in
          inner pp; outer (i + 1) (pp * p)
        end in
      outer 1 p;
    end;
    sum := p * dp.(p) + !sum
  done;
  Printf.printf "%d\n" !sum