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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.

%I A056041 #26 Feb 16 2025 08:32:43
%S A056041 2,3,5,7,23,63,383,2047
%N 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.
%C A056041 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)).
%C A056041 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.
%C A056041 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.
%C A056041 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.
%C A056041 Related to the hereditary base sequences - see cross-reference lines.
%C A056041 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.
%H A056041 R. L. Goodstein, <a href="http://www.jstor.org/stable/2268019">On the Restricted Ordinal Theorem</a>, J. Symb. Logic 9, 33-41, 1944.
%H A056041 L. Kirby, and J. Paris, <a href="https://doi.org/10.1112/blms/14.4.285">Accessible independence results for Peano arithmetic</a>, Bull. London Mathematical Society, 14 (1982), 285-293.
%H A056041 J. Tromp, <a href="https://tromp.github.io/pearls.html">Programming Pearls</a>
%H A056041 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence</a>
%H A056041 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's theorem</a>
%e A056041 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.
%o A056041 Concerning the sequence 2,3,5,7,3*2^402653211 - 1, ... mentioned above, John Tromp write: In Haskell, the sequence is the infinite list
%o A056041 main=mapM_(print.g 2)[0..] where
%o A056041 g b 0=b;g b n=g c(s 0 n-1)where s _ 0=0;s e n=mod n b*c^s 0 e+s(e+1)(div n b);c=b+1
%o A056041 In Ruby, f(n) is defined by
%o A056041 def s(b,e,n)n==0?0:n%b*(b+1)**s(b,0,e)+s(b,e+1,n/b)end
%o A056041 def g(b,n)n==0?b:g(b+1,s(b,0,n)-1)end
%o A056041 def f(n)g(2,n)end
%Y A056041 Cf. A266202, A268687, A268689, A268688.
%Y A056041 Equals A266203 + 2.
%Y A056041 Weak Goodstein sequences: A267647, A267648, A271987, A271988, A271989, A271990, A271991, A137411, A271992, A265034.
%Y A056041 Steps of strong Goodstein sequences: A056004, A057650, A059934, A059935, A059936, A271977.
%Y A056041 Strong Goodstein sequences: A215409, A056193, A266204, A222117, A059933.
%Y A056041 Woodall numbers: A003261.
%K A056041 base,nonn
%O A056041 0,1
%A A056041 _Henry Bottomley_, Aug 04 2000