Alpha Sequence Explorer

What is an Alpha Sequence?

Every time you hit an odd number in the Collatz sequence, two things happen:

1. You compute $3n + 1$ (which always gives an even number)
2. You keep halving until you reach the next odd number

The alpha value is simply how many times you halved. The alpha sequence records this for every odd number you visit, until the orbit reaches 1 (as it has for every number ever tried).

Example: n = 7

Odd number	$3n+1$	Result	Halvings	Alpha
7	$3(7)+1 = 22$	$22 \to 11$	1	1
11	$3(11)+1 = 34$	$34 \to 17$	1	1
17	$3(17)+1 = 52$	$52 \to 26 \to 13$	2	2
13	$3(13)+1 = 40$	$40 \to 20 \to 10 \to 5$	3	3
5	$3(5)+1 = 16$	$16 \to 8 \to 4 \to 2 \to 1$	4	4

Alpha sequence of 7: $[1, 1, 2, 3, 4]$

The alpha sequence is the DNA of the orbit — it completely determines its shape.

Why It Matters

• sum(alphas) = total number of halvings (even steps) in the orbit
• len(alphas) = total number of $3n+1$ steps (odd steps)
• Distinct alphas = the "alphabet" the orbit uses
• Collatz radical = product of distinct alpha values (a name borrowed from the radical of the abc conjecture; my own definition, not a standard one)
• Collatz quality = $\log_2(n) / \log_2(\text{radical})$ — how much "size" $n$ has per unit of orbital complexity

Smooth vs. Rough Orbits

A smooth orbit uses few distinct alpha values — the 3n+1 steps keep hitting similar powers of 2. The smoothest possible orbit has $3n+1 = 2^k$ exactly, giving alpha sequence $[k]$ (a single step!).

A rough orbit uses many distinct alpha values — the 3n+1 steps produce varied results, and the orbit wanders.

The names are borrowed from the abc conjecture, where quality measures how far a sum outruns the primes dividing its terms. Here the borrowing is an analogy in name only: nothing about alpha sequences follows from abc or bears on it, and unlike abc quality (conjecturally bounded) this one is unbounded: $(4^k - 1)/3$ has the one-step alpha sequence $[2k]$.

Try It Yourself

Enter any odd number to see its alpha sequence, radical, and quality:

Enter an odd number:

Alpha Sequence of 27

27→3×27+1 = 82→÷2¹→41α = 1

41→3×41+1 = 124→÷2²→31α = 2

31→3×31+1 = 94→÷2¹→47α = 1

47→3×47+1 = 142→÷2¹→71α = 1

71→3×71+1 = 214→÷2¹→107α = 1

107→3×107+1 = 322→÷2¹→161α = 1

161→3×161+1 = 484→÷2²→121α = 2

121→3×121+1 = 364→÷2²→91α = 2

91→3×91+1 = 274→÷2¹→137α = 1

137→3×137+1 = 412→÷2²→103α = 2

103→3×103+1 = 310→÷2¹→155α = 1

155→3×155+1 = 466→÷2¹→233α = 1

233→3×233+1 = 700→÷2²→175α = 2

175→3×175+1 = 526→÷2¹→263α = 1

263→3×263+1 = 790→÷2¹→395α = 1

395→3×395+1 = 1186→÷2¹→593α = 1

593→3×593+1 = 1780→÷2²→445α = 2

445→3×445+1 = 1336→÷2³→167α = 3

167→3×167+1 = 502→÷2¹→251α = 1

251→3×251+1 = 754→÷2¹→377α = 1

377→3×377+1 = 1132→÷2²→283α = 2

283→3×283+1 = 850→÷2¹→425α = 1

425→3×425+1 = 1276→÷2²→319α = 2

319→3×319+1 = 958→÷2¹→479α = 1

479→3×479+1 = 1438→÷2¹→719α = 1

719→3×719+1 = 2158→÷2¹→1079α = 1

1079→3×1079+1 = 3238→÷2¹→1619α = 1

1619→3×1619+1 = 4858→÷2¹→2429α = 1

2429→3×2429+1 = 7288→÷2³→911α = 3

911→3×911+1 = 2734→÷2¹→1367α = 1

1367→3×1367+1 = 4102→÷2¹→2051α = 1

2051→3×2051+1 = 6154→÷2¹→3077α = 1

3077→3×3077+1 = 9232→÷2⁴→577α = 4

577→3×577+1 = 1732→÷2²→433α = 2

433→3×433+1 = 1300→÷2²→325α = 2

325→3×325+1 = 976→÷2⁴→61α = 4

61→3×61+1 = 184→÷2³→23α = 3

23→3×23+1 = 70→÷2¹→35α = 1

35→3×35+1 = 106→÷2¹→53α = 1

53→3×53+1 = 160→÷2⁵→5α = 5

5→3×5+1 = 16→÷2⁴→1α = 4

Summary

Alpha sequence	[1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4]
Distinct alphas	{1, 2, 3, 4, 5}
Collatz radical	1 × 2 × 3 × 4 × 5 = 120
Collatz quality	log₂(27) / log₂(120) = 0.688
Odd steps (3n+1)	41
Total halvings	70
Rows in the trace above	41
Bits of n	4.8

Numbers to Try

• 3 — Simple: alpha sequence $[1, 4]$, two steps
• 7 — Classic: $[1, 1, 2, 3, 4]$, five odd steps and four distinct alpha values
• 27 — The famous slow number: 41 odd steps, reaches 9232 before descending
• 5461 — Binary $1010101010101$: alpha sequence $[14]$, one step! $3(5461)+1 = 2^{14}$
• 7253 — Highest quality of any odd number under 10000: alpha $[8, 8]$, just two steps
• 1 — Trivial: already at 1, empty alpha sequence