finding the correct time complexity of these codes

I was trying to find time complexity of these 2 codes but i am not sure about my answers .

code1

   int i = 1;
    int count = 0;
   
    while (i < n) {

        for (int j = 0; j < i; j++) {
            count++;
        }
        i *= 2;
    }

I calculated the number of steps in loops and I reached to (log n)^2 but i am not sure about it .

Code2

int k=0;
    for (int i = 2; i <= n; i++) {
        for (int j = 2; j * j <= i; j++) {
            if (i % j == 0) {
                k++;
                break;
            }
        }

    }

and for this one I got ( n * log n)

actually I am new to calculating time complexity and I am not sure about them , could you help me find the correct answer .

Copyright Notice：Content Author:「mas」，Reproduced under the CC 4.0 BY-SA copyright license with a link to the original source and this disclaimer.
Link to original article：https://stackoverflow.com/questions/64295148/finding-the-correct-time-complexity-of-these-codes