A386792 Antihydra, a BB(6) Turing machine (values of a).

8, 12, 18, 27, 40, 60, 90, 135, 202, 303, 454, 681, 1021, 1531, 2296, 3444, 5166, 7749, 11623, 17434, 26151, 39226, 58839, 88258, 132387, 198580, 297870, 446805, 670207, 1005310, 1507965, 2261947, 3392920, 5089380, 7634070, 11451105, 17176657, 25764985, 38647477

Offset: 0

Author

Peter Luschny, Aug 02 2025

Keywords

• nonn

Comments

Let f(a, b) = (3*a / 2, b + 2) if a is even, otherwise ((3*a - 1)/2, b - 1). The Busy Beaver challenge is to find natural numbers n and v, such that after n iterations, starting with f(8, 0), the result is (v, -1).

We consider the sequence to be defined for all values of n, regardless of whether 'b' ever takes the value -1 or not. The sequence records the values of 'a' in this iteration. See A385902 for the values of 'b'.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..5000
• The Busy Beaver Challenge, Antihydra.
• The Busy Beaver Wiki, Antihydra.
• Shawn Ligocki, BB(6) is Hard (Antihydra), July 2024.

Crossrefs

Cf. A385902 (values of b), A028444, A060843.

Programs

Maple

aList := proc(halt) a := 8; u := 0; H := 8;
from 0 to halt do
    q, r := iquo(a, 2), irem(a, 2):
    a, u := a + q, u + r: H := H,a
od: H end: aList(37);

Mathematica

A386792List[n_] := Module[{next},
    next[{a_, u_}] := {a, u} + QuotientRemainder[a, 2];
    NestList[next, {8, 0}, n][[All, 1]]];
A386792List[38] (* Peter Luschny, Aug 05 2025 *)

Python

def A386792List(n) -> list[int]:
    (a, b), H = (8, 0), [8]
    for _ in range(n):
        a, b = map(sum, zip((a, b), divmod(a, 2)))
        H.append(a)
    return H