A349482 Bases where the n-th Goodstein sequence starting in base 3 (instead of base 2) reaches 0.

4, 5, 7, 9, 11, 15, 19, 23, 63, 159, 2047, 10239, 49151, 1048575, 20971519, 402653183, 1180591620717411303423, 233840261972944466912589573234605283144949206876159

Offset

1,1

Comments

a(A056004(n)) lists the bases where the n-th Goodstein sequence starting in base 2 reaches 0. That sequence goes 3, 5, 7, 3*2^402653211 - 1, ...

The Goodstein function is sometimes given as the base where the sequence last has a nonzero value. Following this definition decreases each term in the above sequence by 1.

Like the Goodstein function (which starts in base 2), this sequence appears to grow faster than f_alpha if and only if alpha is smaller than epsilon_0.

As given by the formula below, the sequence continues with a(20,...,26) = 3*2^391 - 1, 4*2^2057 - 1, 5*2^10251 - 1, 3*2^49166 - 1, 4*2^1048594 - 1, 5*20971540 - 1, 3*402653211 - 1.

Links

Table of n, a(n) for n=1..18.

Fine Design, Continue the sequence: 2, 4, 6, _? (BIGNUM BAKEOFF Part 2).

Jonathan F. Waldmann, Goodstein sequences starting in base 3.

Wikipedia, Goodstein's Theorem

Formula

For n<27, i.e. n = a*3^2 + b*3 + c with a, b, c < 3, a(n) = f_2^a (f_1^b (f_0^c (4) ) ) - 1.

Example

For n=6, we write 6 = 2*3.
Then in each step, we increase the base by 1, then subtract 1. This sequence goes
2*3 -> 1*4 + 3 -> 1*5 + 2 -> 1*6 + 1 -> 1*7 + 0 -> 0*8 + 7 ->...-> 0*14 + 1 -> 0*15, so 0 is reached at base 15.

Crossrefs

Cf. A056004.

Sequence in context: A267087 A050560 A204664 * A283554 A035243 A035255

Adjacent sequences: A349479 A349480 A349481 * A349483 A349484 A349485

Keyword

base,nonn

Author

Jonathan F. Waldmann, Nov 19 2021

Status

approved