Usually, the Collatz function is defined by $C(n)=3n+1$ if $n$ is odd, and $C(n)=\frac{n}{2}$ if $n$ is even. The famous conjecture states that some iterate of $C$ evaluated at $n$ is $1$, for any positive integer $n$.
I've spent a lot of time playing around with this problem, and one thing I've noticed is that the "division by 2" step doesn't seem too crucial to whatever mysterious mechanism governs this behavior.
Specifically, we can reformulate the conjecture (equivalently) without division:
Define $C(n)$ to equal $3n+2^k$, where $k$ is the smallest exponent occurring in the base-2 expansion of the (positive integer) input $n$.
Now, no matter which $n$ we start with, the sequence $C(n),C^2(n),\ldots$ eventually reaches a power of 2.
This is essentially the same as the usual formulation, in which we divide by 2 until the "$k$" value is just zero. (We still must reach a power of 2 before getting to 1.)
Perhaps this version is less convenient computationally since the sequence values get larger and larger. On the other hand, the statement seems, at least notationally, more streamlined.
I'm just wondering if there are other apparent advantages (or disadvantages) to stating the conjecture in this way.
Also, I would appreciate any references to this particular formulation.
Well, yes, you can express the Collatz algorithm without division (although fractions look nicer). Here's just one way to do it to generate odd-only sequences:
If $n+1$ is divisible by $4$, multiply by $1.5$, subtract $1$
If $n-1$ is divisible by $8$, multiply by $0.75$, add $1$
Else multiply $n-1$ by $0.25$
Stated as a somewhat clunky function:
$f(n) = \begin{cases} ((n+1)*1.5)-1) & \mbox{if } n+1 \equiv 0 \mbox{ (mod } 4) \\ ((n-1)*0.75)+1) & \mbox{if } n-1 \equiv 0 \mbox{ (mod } 8) \\ (n-1) * 0.25 & \mbox{if } n-1 \equiv 4 \mbox{ (mod } 8)\end{cases}$
Of course, the distinction between decimal fractions and division is nonexistent. However, this reveals the "hidden" bijection - all integers of the form $8n+5: f(n−1)\frac{1}{4}$ - that produces superset odd numbers.
One reason for this alternative statement of the conjecture is clarity. The even numbers may be fruitfully dismissed as ornaments - although an even-only proof is also possible.