Source

This problem is adapted from Long Long OJ. All rights reserved.

Attention

is the hard version of this problem.

The memory limit for this problem is 8 MB.

Problem Description

Define a function:

$$\text{Good}(N)=2N-\sum_{i|N}i$$

That is, $2N$ minus the sum of all divisors of $N$.

Little A wants to know the result of the following expression; please help him calculate it:

$$\text{Good}(l)+\text{Good}(l+1)+\ldots+\text{Good}(r)$$

The answer should be taken modulo $998244353$.

Input Format

A single line containing two positive integers $l, r\ (1 \le l \le r \le 5 \times 10^8)$.

Output Format

A single line containing a non-negative integer representing the answer.

The answer should be taken modulo $998244353$.

Samples

1 3

4

1 2000

710989

1 50000000

10806582

114 514

44867

Sample 3 Explanation

The answer before taking the modulo is $443832427326971$.