Problem:
Score : 600 points

Problem Statement

Find the number of sequences of length K consisting of positive integers such that the product of any two adjacent elements is at most N, modulo 10^9+7.

Constraints

1\leq N\leq 10^9

1 2\leq K\leq 100 (fixed at 21:33 JST)

N and K are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the number of sequences, modulo 10^9+7.

Sample Input 1

3 2

Sample Output 1

5

(1,1), (1,2), (1,3), (2,1), and (3,1) satisfy the condition.

Sample Input 2

10 3

Sample Output 2

147

Sample Input 3

314159265 35

Sample Output 3

457397712

Solution in Python (3.4.3) (judge verdict: Accepted)

import numpy as np

n, k = map(int, input().split())
mod = 10 ** 9 + 7

mx = np.sqrt(n).astype(int)

dp_u = np.ones(mx, np.int64)

N = np.full(mx, n, np.int64)
L = N // np.arange(1, mx + 1)
R = np.maximum(N // np.arange(2, mx + 2), mx)
coef = L - R
dp_d = coef

for _ in range(2, k + 1):
    acc_u = np.cumsum(dp_u) % mod
    acc_d = np.cumsum(dp_d[::-1])[::-1] % mod
    dp_u = (dp_u.sum() + acc_d) % mod
    dp_d = coef * acc_u % mod

ans = (dp_u.sum() + dp_d.sum()) % mod
print(ans)