This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337805 #19 Jul 07 2025 23:42:40
%S A337805 2,7,22,72,427
%N 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.
%C A337805 This sequence and the Busy Beaver (A060843) problem are closely related. Turing machines and the number of steps taken by a Turing machine on an initially blank tape are defined in A060843.
%C A337805 This sequence is computable, while the Busy Beaver problem is uncomputable.
%C A337805 a(n) - 1 <= BB(n), where BB(n) = A060843(n).
%C A337805 a(n) - 1 <= A107668 * 4^(2n), the number of uniquely behaving n-state Turing machines with 2 symbols, by the pigeonhole principle.
%H A337805 Scott Aaronson, <a href="https://www.scottaaronson.com/papers/bb.pdf">The Busy Beaver Frontier</a>, 2020.
%H A337805 Scott Aaronson, <a href="https://www.scottaaronson.com/blog/?p=4916">The Busy Beaver Frontier</a> (blog post)
%e A337805 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.
%Y A337805 Known upper bounds of a(n) - 1 are A028444, A004147, and A141475.
%K A337805 nonn,more
%O A337805 1,1
%A A337805 _Zachary Vance_, Sep 23 2020