cp's OEIS Frontend

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.

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A131957 Busy Beaver sigma variation: maximum number of 1's on the final tape, for a 2-state, 2-symbol Turing machine running on a tape which is initialized with the number n in binary and 0's everywhere else. The machine is started at the rightmost bit in the number n.

Original entry on oeis.org

4, 4, 4, 5, 6, 6, 5, 6, 5, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 6, 8, 8, 7, 8, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 6, 8, 7, 7, 7, 7, 6, 8, 8, 9, 8, 9, 8, 10, 7, 8, 7, 7, 7, 9, 8, 9, 7, 8, 8, 9, 8, 9, 8, 9, 8, 9, 8, 8, 8, 7, 8, 10, 8, 7, 7, 8, 8, 8, 7, 8, 8, 8, 7, 10, 10, 10, 9, 10, 8, 10, 10, 10, 10, 10, 9, 12, 8, 9, 7, 9
Offset: 0

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Author

Bryan Jacobs (bryanjj(AT)gmail.com), Aug 01 2007

Keywords

Examples

			a(5) is the maximum number of 1's on a tape which is initialized as:
..000001010000.... with the machine starting at the rightmost 1.
a(5) = 6, with the machine:
A0-> 1BL
A1-> 1AR
B0-> 1*L
B1-> 1AL
		

Crossrefs

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.

Original entry on oeis.org

2, 7, 22, 72, 427
Offset: 1

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Author

Zachary Vance, Sep 23 2020

Keywords

Comments

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.
This sequence is computable, while the Busy Beaver problem is uncomputable.
a(n) - 1 <= BB(n), where BB(n) = A060843(n).
a(n) - 1 <= A107668 * 4^(2n), the number of uniquely behaving n-state Turing machines with 2 symbols, by the pigeonhole principle.

Examples

			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.

A386909 Iterates of g(x), starting at x = 0 until g(x) == 2 (mod 3), where g(x) = (5*x + 18)/3 if x == 0 (mod 3) and g(x) = (5*x + 22)/3 if x == 1 (mod 3).

Original entry on oeis.org

0, 6, 16, 34, 64, 114, 196, 334, 564, 946, 1584, 2646, 4416, 7366, 12284
Offset: 0

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Author

Paolo Xausa, Aug 07 2025

Keywords

Comments

These iterates correspond to the computation carried out by the 5-state Marxen-Buntrock Turing machine, which halts after 47176870 = BB(5) = A060843(5) steps (see Aaronson link).

Crossrefs

Cf. A060843.

Programs

  • Mathematica
    NestWhileList[(5*# + If[Mod[#, 3] == 0, 18, 22])/3 &, 0, Mod[#, 3] < 2 &]
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