Logarithmic Summations and Discrete Loops - Ceiling and Floor Functions

Reading the so called paper "Discrete Loops and Worst Case Performance" (by Dr. Johann Blieberger), I came across some interesting summations:

Useful here:

for ( i = 1; i <= n ; i ++ ) {
    for (j = 1; j <= i ; j *= 2) {
        // Some constant time C instructions.
    }
}

Because:

Likewise:

Applicable here:

for ( i = 1; i <= n ; i ++ ) {
    for (j = 1; j < i ; j *= 2) {
        // Some constant time C instructions.
    }
}

Because:

And more generally:

for ( i = 1; i <= n ; i ++ ) {
    for (j = 1; j <= i ; j *= a) {
        // Some constant time C instructions.
    }
}

We can proceed like the following:

Further:

for ( i = 1; i <= n ; i ++ ) {
    for (j = 1; j < i ; j *= a) {
        // Some constant time C instructions.
    }
}

With:

Thus, while I tried to refine the results of some nested loops running time analysis, I had to face a particular summation, using both logarithm and ceiling functions, but this time, as an exponent. I finally deduced a general solution for the range [1, n]:

And similarly:

I hope this was useful to you. Enjoy!