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I have a non-zero integer $x \in \mathbb{N}$, and I want to represent it as

$$ x = 2^N \cdot Q $$

where $N,Q \in \mathbb{N_0}$, and Q is an odd number. That means, $x$ is separated into an part that is a power-of-2, and an odd part.

Question: What is an explicit formular, using elementary operations, for $N(x)$ and $Q(x)$?


Special cases:

The two trivial special cases:

  • If $x$ is odd: $N(x)=0$, $Q(x)=x$
  • If $x$ is a power-of-two: $N(x)=\log_2(x)$, $Q(x)=1$

Added later: Elementary function:

I would like to get functions in the spirit of the $max$-function in terms summation, multiplication, and absolute, or as a limit, or any other elementary operations.

The solution by gammatester, while clearly answering the question about representation, uses conditions which i consider as not elementary. Is it possible to represent it in an simpler, more elementary way?

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  • $\begingroup$ Is $0$ contained in the naturals for you? Otherwise this doesn't work. No odd number can be expressed as a multiple of 2 since even*odd=even $\endgroup$ – Aaron Zolotor Oct 28 '18 at 14:02
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    $\begingroup$ What do you mean with find? An explicit formula? For $x\ne 0$ your $N(x)$ is the valuation $\nu_2(x)$ $\endgroup$ – gammatester Oct 28 '18 at 14:04
  • $\begingroup$ Thank you, i clarified. Gammatester, the connection to valuation is very interesting. I think you can convert your comment into an answer, right? $\endgroup$ – NicoDean Oct 28 '18 at 14:20
  • $\begingroup$ @LeeMosher please try again, i initially had a wrong link, but fixed it. $\endgroup$ – NicoDean Oct 28 '18 at 14:55
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    $\begingroup$ Does this count: $N(x)=\lfloor \cos^2(x \pi/2) \rfloor + \lfloor \cos^2(x\pi/4) \rfloor +\cdots + \lfloor \cos^2(x \pi/2^x) \rfloor.$ $\endgroup$ – Marco Oct 28 '18 at 15:06
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If $x=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ is the prime factorization of $x$, one can represent $\alpha_i(x)$ as

$$\alpha_i(x)=\lfloor \cos^2(x \pi/p_i) \rfloor + \lfloor \cos^2(x\pi/p_i^2) \rfloor +\cdots + \lfloor \cos^2(x \pi/p_i^x) \rfloor.$$

So in particular

$$N(x)=\lfloor \cos^2(x \pi/2) \rfloor + \lfloor \cos^2(x\pi/4) \rfloor +\cdots + \lfloor \cos^2(x \pi/2^x) \rfloor,$$ and then $Q(x)=x/2^{N(x)}$.

Update: a more elementary function. Let $N_K(x)$ be defined by $$N_k(x)=\left \lfloor \frac{x}{2^k} \right \rfloor - \left \lfloor \frac{x-1}{2^k} \right \rfloor.$$ Then $N_k(x)=1$ if and only if the positive integer $x$ is divisibly by $2^k$. To see this, let $x=2^ky+r$, where $0\leq r<2^k$. If $r\neq 0$, then $\lfloor x/2^k \rfloor =y$ and $\lfloor (x-1)/2^k \rfloor =y$, and so $N_k(x)=0$. While if $r=0$, then $\lfloor (x-1)/2^k \rfloor =y-1$, and so $N_k(x)=1$ in this case.

It follows that $$N(x)=\sum_{k=2}^\infty N_k(x)=\sum_{k=2}^x \left \lfloor \frac{x}{2^k} \right \rfloor - \left \lfloor \frac{x-1}{2^k} \right \rfloor,$$ since the largest possible value for $k$ such that $2^k$ divides $x$ cannot exceed $x$ ($2^x \geq x$ for all integers).

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  • $\begingroup$ Thank you. How about you add the specific formular for $N(x)$ from your comment, in order to fully answer my question. I will upvote and probably accept your answer (if nothing more elementary appears in near future, which i doubt). This method is really cool - thanks. $\endgroup$ – NicoDean Oct 28 '18 at 17:54
  • $\begingroup$ @NicoDean thanks I added a more elementary function. Please feel free to edit my post by adding how you can express the floor function. $\endgroup$ – Marco Oct 29 '18 at 11:33
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This is kind of an explicit formula for $x\ne 0$:

$$N(x) = \mathrm{max}\{n \in \mathbb{N} : 2^n \mid x\} \quad \text{and} \quad Q(x)=x / N(x).$$ Algorithmically you repeatedly divide $x$ by $2$ and count the number $N(x)$ of divisions until you get a non-zero remainder $Q(x).$

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  • $\begingroup$ Thank you. Your answer is clearly interesting. However, would it possible to write the representation in terms of more elementary functions? I updated the question to cover this. Thanks! $\endgroup$ – NicoDean Oct 28 '18 at 14:47
  • $\begingroup$ I doubt that you can find such a function in your given sets. At least you additionally need a sort of $\mathrm{mod}$ function, and even then the function is wildly jumping, look and $x=16,17,18$. $\endgroup$ – gammatester Oct 28 '18 at 15:04
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I think this is possible.

In fact I think it's possible to do this for every single function $f : \mathbb N \to \mathbb R$.

First, write $f$ as a limit of a sequence of piecewise linear functions $$f = \lim_{n \to \infty} F_n $$ where for each $i=2,..,n$ the restriction of $F_n$ to $[i-1,i]$ is the unique linear function with $F_n(i-1)=f(i-1)$ and $F_n(i)=f(i)$, the restriction of $F_n$ to $[-\infty,1]$ is constant, and the restriction of $F_n$ to $[n,\infty]$ is constant.

Next, every piecewise linear function $F(x)$ can be written as a finite formula involving only first degree functions $ax+b$ and max.

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