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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
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
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
KEYWORD
nonn,more
AUTHOR
Zachary Vance, Sep 23 2020
STATUS
approved