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

%I A349482 #7 Dec 15 2021 02:35:41
%S A349482 4,5,7,9,11,15,19,23,63,159,2047,10239,49151,1048575,20971519,
%T A349482 402653183,1180591620717411303423,
%U A349482 233840261972944466912589573234605283144949206876159
%N A349482 Bases where the n-th Goodstein sequence starting in base 3 (instead of base 2) reaches 0.
%C A349482 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, ...
%C A349482 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.
%C A349482 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.
%C A349482 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.
%H A349482 Fine Design, <a href="https://youtu.be/hm3iOoTQCLA">Continue the sequence: 2, 4, 6, _? (BIGNUM BAKEOFF Part 2)</a>.
%H A349482 Jonathan F. Waldmann, <a href="https://docs.google.com/spreadsheets/d/14q74Wez0aGPIlA7W0141-m6JZEVE9jx44784PctbJRc/edit?usp=sharing">Goodstein sequences starting in base 3</a>.
%H A349482 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goodstein%27s_theorem">Goodstein's Theorem</a>
%F A349482 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.
%e A349482 For n=6, we write 6 = 2*3.
%e A349482 Then in each step, we increase the base by 1, then subtract 1. This sequence goes
%e A349482 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.
%Y A349482 Cf. A056004.
%K A349482 base,nonn
%O A349482 1,1
%A A349482 _Jonathan F. Waldmann_, Nov 19 2021