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Given the following code:

void aux(int n){
   while (n > 0) {
   printf("%d", n);
   n--;
  }
}
long f1(int n)
{
  long sum = 0;
  for (int i = 1; i * i <= n; i++) {
    for (int j = 1; j <= i; j++) {
      aux(i * j);
      sum += i * j;
     }
  }
 return sum;
}

I have calculated the time complexity of aux and I found it is $\Theta(n)$. Then I tried to find the time complexity of f1 and got the following expression:

$$ \sum_{j=1}^1 j + 2\sum_{j=1}^2j + 3\sum_{j=1}^3 j +...+ \sqrt{n} \sum_{j=1}^{\sqrt{n}} j $$

The right answer here is $\Theta(n^2)$ but I'm really not sure why that is the case. Can anyone explain if the expression I've got is correct, and if so why it is equal to the time complexity above?

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1 Answer 1

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Let $n=m^2$.

$$ \sum_{j=1}^1 j + 2\sum_{j=1}^2j + 3\sum_{j=1}^3 j +...+ m \sum_{j=1}^{m} j \\=1 + 2\cdot3 + 3\cdot6 +...+ m \frac{m(m+1)}2\\\sim\frac{m^4}8=\Theta(m^4)=\Theta(n^2) $$

by Faulhaber.

More precisely,

$$\frac{3m^4+10m^3+9m^2+2m}{24}=\frac{m(3m+1)(m+1)(m+2)}{24}.$$

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