%I A056041
%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 (tromp(AT)cwi.nl), 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) 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.
%D A056041 Goodstein, R., On the restricted ordinal theorem, Journal of Symbolic 
               Logic, 9 (1944), 33-41.
%D A056041 Kirby, L. and Paris, J., Accessible independence results for Peano arithmetic, 
               Bull. London. Math. Soc., 14 (1982), 285-93.
%H A056041 J. Tromp, <a href="http://homepages.cwi.nl/~tromp/pearls.html">Programming 
               Pearls</a>
%H A056041 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GoodsteinSequence.html">Goodstein Sequence</a>
%H A056041 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein'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. Goodstein sequences: A056041 A056004 A059934 A057650 A056193 A059933 
               A059935 A059936; Woodall numbers: A003261.
%Y A056041 Sequence in context: A068710 A120805 A096265 this_sequence A083017 A052087 
               A006510
%Y A056041 Adjacent sequences: A056038 A056039 A056040 this_sequence A056042 A056043 
               A056044
%K A056041 base,nonn
%O A056041 0,1
%A A056041 Henry Bottomley (se16(AT)btinternet.com), Aug 04 2000