

A337805


Lazy Beaver Problem: a(n) is the smallest positive number of steps a(n) such that no nstate Turing machine halts in exactly a(n) steps on an initially blank tape.


0




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 nstate 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 2state 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



