Is there a name for this function or a concept similar to it?

Question

I'm wondering if anyone has heard or seen a function that looks or behaves like this one.

$$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}$$

'b' is a constant we choose. I've been calling this the "Proportional Power Ratio of x, base b", but I wonder if it goes by another name.

Here is a link to a Desmos graph where you can play with it and vary the base b. https://www.desmos.com/calculator/obedfxwb6l

Essentially the values get "smeared out' between powers of b.

Here is a graph of f(x) over integers in the interval [2, 256] where b is equal to 2:

f(x) over integers in the interval [2, 256] where b is equal to 2

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If you're more interested in the application of this, I posted an article on Medium about how I've been using this function to explore the Collatz Conjecture:

https://medium.com/@vardthomas/the-collatz-conjecture-a-new-perspective-on-an-old-problem-f4bca7ff675a

Answer 1

$\def\floor{\operatorname{floor}}$

Let $\#x$ denote the number of digits that are needed to represent $x\in\Bbb N$ to base $b$, then

$$\#x = 1 + \floor (\log_b x)$$

and $b^{\#x-1}$ is the natural number $10\cdots0$ in base $b$ that has the same number of digits like $x$ in base $b$. Then

$$\begin{align} f(x) &= \frac{x-b^{\floor(\log_b x)}}{b^{\floor(\log_b x) + 1}-b^{\floor(\log_b x)}} \\ &= \frac {x-b^{\#x-1}}{b^{\#x} - b^{\#x-1}} \\ &= \frac 1{b-1} \cdot \left(\frac{x}{b^{\#x-1}} -1\right) \\ &\stackrel{(*)}= \frac{x}{b^{\#x-1}} -1 \tag 1 \end{align}$$

where in $(*)$ we ignore the constant factor of $1/(b-1)$.

The remaining formula $(1)$ then can be statet as:

Take a natural number and insert a decimal point after the most significant digit, and then subtract$\ 1$.

For example is base 3, we'd have: 21001201 → 2.1001201 → 1.1001201

One could also rephrase it as

A relative measure of how far away a number is from the smallest number with the same number of digits.

Dunno for what's that good for, though...