[seqfan] Goodstein sequence offsets

Nick Matteo kundor at kundor.org
Tue Aug 20 20:35:24 CEST 2019

The OEIS is currently inconsistent with the starting offset of
Goodstein sequences.

By my count, there are 49 sequences in the OEIS involving these sequences.
(A full list is at the end of this email.)

Ten of them have offset 2, or (for the sequences of the form "kth step in
Goodstein sequences") are described consistently with such an offset. (This has
the convenient feature that term a(n) is expressed in base n.)

34 sequences use an offset of 0, so that term a(n) is expressed in base n+2.

One sequence uses offset 1, and the remaining four are offset-agnostic.

Personally, I like offset 2, but the literature seems to be in favor of offset
0.  Goodstein's own paper "On the restricted ordinal theorem" (1944) starts
from any number n_0 and allows any non-decreasing sequence p_r of bases,
starting with p_0 >= 2; then n_{r+1} is obtained by bumping the hereditary base
p_r expression of n_r to base p_{r+1}, and subtracting one.  ("Goodstein
sequences" use p_r = r + 2 as the bases.)

Kirby and Paris "Accessible independence results for Peano arithmetic" (1982)
also use offset 0, defining m_0 = m for the initial term, then m_n =
G_{n+1}(m_{n-1}), where G_k(m) expresses m in hereditary base k and bumps to
base k+1 (and subtracts one).

Other sources, including Hrbacek & Jech _Introduction to Set Theory_ (1999, pp.
125-127), Hodgson "Herculean or Sisyphean Tasks" (Newsletter of the EMS, 2004),
the Googology wiki https://googology.wikia.org/wiki/Goodstein_sequence, and
MathWorld http://mathworld.wolfram.com/GoodsteinSequence.html, agree with
offset 0.

A few sources (notably Wikipedia
also Caicedo "Goodstein's function" (Revista Colombiana de Matemáticas, 2007)
prefer offset 1.

I couldn't find any support for offset 2.

Therefore, I propose changing the sequences currently using offset 2 to be
offset 0 instead.  If no one objects I'll submit revisions to the sequences (I
have b-files to add also).

Nick Matteo

Offset 2:
A056193 "Goodstein sequence with a(2) = 4" (Henry Bottomley)
A222117 "Goodstein sequence starting with 15" (Reinhard Zumkeller)
A059933 "Goodstein sequence with a(2)=16" (HB)
A211378 "Goodstein sequence starting with 19" (RZ)
A137411 "Weak Goodstein sequence starting at 11" (myself)
A056041 Base where weak Goodstein sequence reaches 0 (HB)
A057650 "Second step in Goodstein sequences, i.e., g(4) if g(2)=n" (HB)
A059934 "Third step in Goodstein sequences, i.e., g(5) if g(2)=n" (HB)
A059935 "Fourth step in Goodstein sequences, i.e., g(6) if g(2)=n" (HB)
A059936 "Fifth step in Goodstein sequences, i.e., g(7) if g(2)=n" (HB)

Offset 0:
A215409 G_n(3) (Jonathan Sondow)

All the following are by Natan Arie' Consigli:
A266204, A266205, A271554, A271555, A271556, A271557, A271558, A271559,
A271560, A271561: G_n(5) through G_n(14)
A271562 G_n(17)
A271975 G_n(18)
A217976 G_n(20)
A266201 G_n(n)
A271977, A271978, A271979, A271985, A271986: Sixth through tenth steps
A267647, A267648, A271987, A271988, A271989, A271990, A271991: weak Goodstein
sequences g_n(4) through g_n(10)
A271992 g_n(16)
A266202 g_n(n)
A266203 "Number of steps k to make g_k(n) converge to zero" (= A056041(n) - 2)
A268688, A268689 Last or first index, respectively, of the maximum value in
the weak Goodstein sequence for n

A265034 "Weak Goodstein sequence beginning with 266" (N.J.A. Sloane)
A296441 "Array A(n,k) = G_k(n)" (Iain Fox)

Offset 1:
A222113 "Goodstein sequence starting with a(1) = 16" (RZ)

A056004 "Initial step in Goodstein sequences" (HB)
A222112 "Initial step in Goodstein sequences" (when subtracting first) (RZ)
A268687 "a(n) = MAX(g_k(n))" (NAC)
A300404 "Smallest integer k such that the largest term in the Goodstein
sequence starting at k is > n" (Felix Fröhlich)

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