Sum of GCD-Power Series with Squarefree Filter

You are given two integers n and k. Compute the following sum modulo 998244353:

\[ S = \sum_{i=1}^{n}\sum_{j=1}^{n} (i+j)^k\, f(\gcd(i,j))\, \gcd(i,j), \] where \(\gcd(i,j)\) denotes the greatest common divisor of \(i\) and \(j\).

The function \(f\) is defined as follows:

• If the argument has a square factor (i.e. there exists an integer \(k > 1\) such that \(k^2\) divides the number), then \(f(\text{number}) = 0\).
• Otherwise, \(f(\text{number}) = 1\).

In other words, for any positive integer \(m\), \(f(m)=0\) if \(m\) is not squarefree, and otherwise \(f(m)=1\).

Output the sum \(S\) modulo \(998244353\).

inputFormat

The input consists of a single line with two integers n and k (1 ≤ n and k ≥ 0).

outputFormat

Output a single integer: the value of \(S\) modulo \(998244353\).

sample

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16