Core Definitions

This page collects the formal definitions used throughout the site. Each definition includes a worked example. The notation follows Paper 1 (dropping) conventions; see the Terminology Map for equivalences with Paper 2 (stopping). None of these objects is new: the dropping time is the stopping time of Terras (1976) counted in single steps, a dropping set is the set of integers with a given stopping time, and the Syracuse map is classical. Only the names differ.

Collatz Step

Definition 1 (Collatz Step). The Collatz function $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ is defined by

$$f(n) = \begin{cases} n/2 & \text{if } n \text{ is even} \\ 3n+1 & \text{if } n \text{ is odd} \end{cases}$$

Example. $f(7) = 3(7)+1 = 22$ (odd rule), then $f(22) = 22/2 = 11$ (even rule).

Orbit

Definition 2 (Orbit). The orbit of $n$ is the sequence $n,\; f(n),\; f^2(n),\; \ldots$ obtained by iterating $f$. From 1 it repeats $1 \to 4 \to 2 \to 1$ forever, so an orbit that reaches 1 is written up to its first 1:

$$\text{orbit}(n) = [n,\; f(n),\; f^2(n),\; \ldots,\; 1]$$

Whether every orbit reaches 1 is the Collatz conjecture.

Example. The orbit of 6 is $[6, 3, 10, 5, 16, 8, 4, 2, 1]$.

Dropping Time

Definition 3 (Dropping Time). The dropping time of $n > 1$ is the smallest $k \geq 1$ such that $f^k(n) < n$, if there is one:

$$\text{drop}(n) = \min\{k \geq 1 : f^k(n) < n\}$$

This is the stopping time of the literature (Terras, 1976, who counts an odd step and the halving after it as one step) and of Paper 2. The OEIS also calls it the dropping time (A060445).

Example. For $n = 5$: $f(5) = 16$, $f^2(5) = 8$, $f^3(5) = 4$. Since $4 < 5$ and this is the first value below 5, the dropping time is 3.

Dropping Destination

Definition 4 (Dropping Destination). The dropping destination of $n$ is the first iterate that falls below $n$:

$$\text{dest}(n) = f^{\text{drop}(n)}(n)$$

Example. The dropping destination of 5 is $f^3(5) = 4$.

Dropping Orbit

Definition 5 (Dropping Orbit). The dropping orbit of $n$ is the segment of the orbit from $n$ up to (but not including) the dropping destination $d$:

$$\text{dorbit}(n) = [n,\; f(n),\; \ldots,\; 2d]$$

It includes $n$ and excludes the destination. Its length equals the dropping time.

Example. The dropping orbit of 5 is $[5, 16, 8]$ (length 3). The destination $4$ is excluded.

For $n = 3$: orbit proceeds $3 \to 10 \to 5 \to 16 \to 8 \to 4$, and the destination is $2 < 3$. The dropping orbit is $[3, 10, 5, 16, 8, 4]$ (length 6).

Dropping Set

Definition 6 (Dropping Set). For each positive integer $k$, the dropping set $\text{Dset}_k$ is defined as:

$$\text{Dset}_k = \{n \in \mathbb{Z} : n > 1,\; f^k(n) < n, \text{ and } f^j(n) \geq n \text{ for all } 0 < j < k\}$$

That is: the set of all integers whose first drop below themselves occurs at exactly step $k$.

Note on existence

This is a definition, not a claim. We do not assert that every integer greater than 1 belongs to some $\text{Dset}_k$ — that assertion would be equivalent to the Collatz conjecture. We define $\text{Dset}_k$ as the set of integers with dropping time $k$, and the results on this site describe properties of these sets. An integer with no finite dropping time would simply not belong to any $\text{Dset}_k$.

Example.

• $\text{Dset}_1$ = all even numbers $\{2, 4, 6, 8, \ldots\}$, since one halving gives $n/2 < n$.
• $\text{Dset}_3 = \{5, 9, 13, 17, \ldots\} = \{n > 1 : n \equiv 1 \pmod{4}\}$.

Both examples are arithmetic progressions. For a general dropping set the Affine Orbit Structure gives this much: the pattern of odd and even steps in the first $k$ steps depends only on $n$ modulo a power of 2, and when a pattern's multiplier $3^s/2^{k-s}$ (with $s$ odd steps) first falls below 1 at step $k$, every large enough number with that pattern has dropping time exactly $k$. That every such number greater than 1 does, so that each $\text{Dset}_k$ is exactly a union of arithmetic progressions, is Terras's coefficient stopping time conjecture (so named by Lagarias, 1985). It is open, it would rule out nontrivial cycles, and it has been verified for all $n$ from 2 to $2.8 \times 10^{19}$ (Rozier and Terracol, 2026).

Orbital Oddity

Definition 7 (Orbital Oddity). The orbital oddity of $n$ is the count of odd numbers in its dropping orbit. Within the verified range above, all members of $\text{Dset}_k$ share the same orbital oddity $s$; that they always do would follow from the coefficient stopping time conjecture.

Example. The orbital oddity of 3 is 2. Its dropping orbit is $[3, 10, 5, 16, 8, 4]$, which contains two odd values: 3 and 5.

Syracuse Map

Definition 8 (Syracuse Map). The Syracuse map $S : \mathbb{Z}_{\text{odd} }^+ \to \mathbb{Z}_{\text{odd} }^+$ compresses each $3n+1$ step and all subsequent halvings into a single operation:

$$S(n) = \frac{3n+1}{2^{v_2(3n+1)} }$$

where $v_2(m)$ is the 2-adic valuation of $m$ (the largest power of 2 dividing $m$).

Example. $S(7) = \frac{3(7)+1}{2^{v_2(22)} } = \frac{22}{2^1} = 11$, since $22 = 2 \cdot 11$.

Alpha Value

Definition 9 (Alpha Value). For odd $n$, the alpha value is the 2-adic valuation of $3n+1$:

$$\alpha(n) = v_2(3n+1)$$

This counts how many halvings follow the $3n+1$ step in the Syracuse map.

Example. $\alpha(5) = v_2(16) = 4$, since $3(5)+1 = 16 = 2^4$.

Alpha Sequence

Definition 10 (Alpha Sequence). The alpha sequence of an odd number $n$ is the sequence of alpha values along its Syracuse orbit until reaching 1:

$$\alpha\text{-seq}(n) = [\alpha(n),\; \alpha(S(n)),\; \alpha(S^2(n)),\; \ldots]$$

The sequence terminates if and when the Syracuse orbit reaches 1.

Example. The alpha sequence of 3 is $[1, 4]$:

• $S(3) = (3 \cdot 3 + 1)/2^1 = 5$, so $\alpha(3) = 1$
• $S(5) = (3 \cdot 5 + 1)/2^4 = 1$, so $\alpha(5) = 4$

The alpha sequence of 7 is $[1, 1, 2, 3, 4]$.