Time Complexity of Loop with Powers

What is the time complexity of the below function?

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void fun(int n, int k)
{
    for (int i = 1; i <= n; i++)
    {
        int p = pow(i, k);
        for (int j = 1; j <= p; j++)
        {
            // Some O(1) work
        }
    }
}
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Time complexity of above function can be written as 1k + 2k + 3k + … n1k.

Let us try few examples: 

k=1
Sum = 1 + 2 + 3 ... n 
    = n(n+1)/2 
    = n2/2 + n/2

k=2
Sum = 12 + 22 + 32 + ... n12.
    = n(n+1)(2n+1)/6
    = n3/3 + n2/2 + n/6

k=3
Sum = 13 + 23 + 33 + ... n13.
    = n2(n+1)2/4
    = n4/4 + n3/2 + n2/4     

In general, asymptotic value can be written as (nk+1)/(k+1) + Θ(nk)
If  n>=k then the time complexity will be considered in O((nk+1)/(k+1)) and if n<k, then the time complexity will be considered as  in the O(nk)

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