A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

2, 3, 5, 7, 23, 63, 383, 2047

Offset: 0

Author

Henry Bottomley, Aug 04 2000

Keywords

• base
• nonn

Comments

a(8)=3*2^(3*2^27+27)-1 which is more than 10^(10^8) and equal to the final base of the Goodstein sequence starting with g(2)=4; indeed, apart from the initial term, the sequence starting with b(2)=8 is identical to the Goodstein sequence starting with g(2)=4. The initial terms of a(n) [2, 3, 5 and 7] are equal to the initial terms of the equivalent final bases of Goodstein sequences starting at the same points. a(9)=2^(2^(2^70+70)+2^70+70)-1 which is more than 10^(10^(10^20)).

It appears that if n is even then a(n) is one less than three times a power of two, while if n is odd then a(n) is one less than a power of two.

Comment from John Tromp, Dec 02 2004: The sequence 2,3,5,7,3*2^402653211 - 1, ... gives the final base of the Goodstein sequence starting with n. This is an example of a very rapidly growing function that is total (i.e. defined on any input), although this fact is not provable in first-order Peano Arithmetic. See the links for definitions. This grows even faster than the Friedman sequence described in the Comments to A014221.

In fact there are two related sequences: (i) The Goodstein function l(n) = number of steps for the Goodstein sequence to reach 0 when started with initial term n >= 0: 0, 1, 3, 5, 3*2^402653211 - 3, ...; and (ii) the same sequence + 2: 2, 3, 5, 7, 3*2^402653211 - 1, ..., which is the final base reached. Both grow too rapidly to have their own entries in the database.

Related to the hereditary base sequences - see cross-reference lines.

This sequence gives the final base of the weak Goodstein sequence starting with n; compare A266203, the length of the weak Goodstein sequence. a(n) = A266203(n) + 2.

Examples

			a(3)=7 because starting with b(2)=3=11 base 2, we get b(3)=11-1 base 3=10 base 3=3, b(4)=10-1 base 4=3, b(5)=3-1 base 5=2, b(6)=2-1 base 6=1 and b(7)=1-1 base 7=0.
		

Links

• R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
• L. Kirby, and J. Paris, Accessible independence results for Peano arithmetic, Bull. London Mathematical Society, 14 (1982), 285-293.
• J. Tromp, Programming Pearls
• Eric Weisstein's World of Mathematics, Goodstein Sequence
• Wikipedia, Goodstein's theorem

Crossrefs

Cf. A266202, A268687, A268689, A268688.

Equals A266203 + 2.

Weak Goodstein sequences: A267647, A267648, A271987, A271988, A271989, A271990, A271991, A137411, A271992, A265034.

Steps of strong Goodstein sequences: A056004, A057650, A059934, A059935, A059936, A271977.

Strong Goodstein sequences: A215409, A056193, A266204, A222117, A059933.

Woodall numbers: A003261.