Counting Prefix GCD Sequences

Given two positive integers n and m, consider an integer sequence a of length m where each element a_i satisfies a_i \in [1,n]. Define another sequence b as the prefix gcd sequence of a, that is,

[ b_1=a_1, \quad b_i=\gcd(a_1, a_2, \dots, a_i) \quad (2 \le i \le m). ]

Note that for any valid b generated in this way, we have b_i+1 \mid b_i for \(1 \le i .

For given positive integers n and m, define a function \(f(n, m, k)\) as the number of distinct sequences b (constructed as above from some sequence a) in which the number of indices \(i\) (with \(1 \le i < m\)) that satisfy b_i = b_i+1 is at most \(k\). In other words, if we denote by \(E\) the count of adjacent equal pairs in the sequence b, we require \(E \le k\).

Your task is to compute the following sum modulo \(2^{32}\):

[ S = \sum_{i=1}^{n} ; \sum_{j=1}^{m} ; \sum_{k=0}^{j-1} f\Bigl(\Bigl\lfloor \frac{n}{i}\Bigr\rfloor,; j,; k\Bigr). ]

Note: All formulas must be interpreted in \(\LaTeX\) format.

inputFormat

The input consists of a single line containing two positive integers n and m.

\(1 \le n, m \le \text{(small enough for a DP solution)}\)

outputFormat

Output a single integer, the value of \(S\) modulo \(2^{32}\).

sample

3 2

14