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A337805 Lazy Beaver Problem: a(n) is the smallest positive number of steps a(n) such that no n-state Turing machine halts in exactly a(n) steps on an initially blank tape. 0
2, 7, 22, 72, 427 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This sequence and the Busy Beaver (A028444) problem are closely related. Turing machines and the number of steps taken by a Turing machine on an initially blank tape are defined in A028444.

This sequence is computable, while the Busy Beaver problem is noncomputable.

a(n) - 1 <= BB(n), where BB(n) = A028444(n).

a(n) - 1 <= (4n+1)^(2n), the number of n-state Turing machines with 2 symbols, by the pigeonhole principle. (4n+1)^(2n) is nearly A141475 (slightly different formalisms are used).

LINKS

Table of n, a(n) for n=1..5.

Scott Aaronson, The Busy Beaver Frontier, 2020.

Scott Aaronson, The Busy Beaver Frontier (blog post)

EXAMPLE

For n = 2, there exist 2-state Turing machines which halt in exactly {1, 2, 3, 4, 5, 6} steps (and for no other number of steps) given an initially empty input tape. a(2) = 7 is defined as the lowest positive integer not present in that set of possible step lengths.

CROSSREFS

Known upper bounds of a(n) - 1 are A028444, A004147, and A141475.

Sequence in context: A292230 A162770 A116387 * A294006 A322573 A294007

Adjacent sequences:  A337802 A337803 A337804 * A337806 A337807 A337808

KEYWORD

nonn,more

AUTHOR

Zachary Vance, Sep 23 2020

STATUS

approved

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Last modified April 21 12:24 EDT 2021. Contains 343150 sequences. (Running on oeis4.)