Search: id:A059933
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%I A059933
%S A059933 16,7625597484986,50973998591214355139406377,
%T A059933 19916489515870532960258562190639398471599239042185934648024761145811,
%U A059933 5103702287864892035208610181878203902270504134895451401860454182513968464023205038690962121196797
%N A059933 Goodstein sequence with a(2)=16: to calculate a(n+1), write a(n) in the 
               hereditary representation base n, then bump the base to n+1, then 
               subtract 1.
%C A059933 Goodstein's theorem shows that such a sequence is finite (i.e. it eventually 
               stablizes and then decreases by 1 in each step until it reaches 0) 
               for any starting point of a(2). In this case of a(2)=16, there seems 
               little possibility of describing how incredibly large n must be for 
               a(n)=0.
%D A059933 Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 
               9, 33-41, 1944
%e A059933 a(2) = 16 = 2^(2^2) so a(3) = 3^(3^3)-1 = 7625597484986.
%e A059933 So a(3) = 2*3^(2*3^2 + 2*3 + 2) + 2*3^(2*3^2 + 2*3 + 1) + 2*3^(2*3^2 
               + 2*3) + 2*3^(2*3^2 + 1*3 + 2) + 2*3^(2*3^2 + 1*3 + 1) + 2*3^(2*3^2 
               + 1*3) + 2*3^(2*3^2 + 2) + 2*3^(2*3^2 + 1) + 2*3^(2*3^2) + 2*3^(3^2 
               + 2*3 + 2) + 2*3^(3^2 + 2*3 + 1) + 2*3^(3^2 + 2*3) + 2*3^(3^2 + 1*3 
               + 2) + 2*3^(3^2 + 1*3 + 1) + 2*3^(3^2 + 1*3) + 2*3^(3^2 + 2) + 2*3^(3^2 
               + 1) + 2*3^(3^2) + 2*3^(2*3 + 2) + 2*3^(2*3 + 1) + 2*3^(2*3) + 2*3^(1*3 
               + 2) + 2*3^(1*3 + 1) + 2*3^(1*3) + 2*3^(2) + 2*3^(1) + 2,
%e A059933 leading to a(4) = 2*4^(2*4^2 + 2*4 + 2) + 2*4^(2*4^2 + 2*4 + 1) + 2*4^(2*4^2 
               + 2*4) + 2*4^(2*4^2 + 1*4 + 2) + 2*4^(2*4^2 + 1*4 + 1) + 2*4^(2*4^2 
               + 1*4) + 2*4^(2*4^2 + 2) + 2*4^(2*4^2 + 1) + 2*4^(2*4^2) + 2*4^(4^2 
               + 2*4 + 2) + 2*4^(4^2 + 2*4 + 1) + 2*4^(4^2 + 2*4) + 2*4^(4^2 + 1*4 
               + 2) + 2*4^(4^2 + 1*4 + 1) + 2*4^(4^2 + 1*4) + 2*4^(4^2 + 2) + 2*4^(4^2 
               + 1) + 2*4^(4^2) + 2*4^(2*4 + 2) + 2*4^(2*4 + 1) + 2*4^(2*4) + 2*4^(1*4 
               + 2) + 2*4^(1*4 + 1) + 2*4^(1*4) + 2*4^(2) + 2*4^(1) + 1 = 2*(4^32 
               + 4^16 + 1)*(4^8 + 4^4 + 1)*(4^2 + 4*1)-1 = 50973998591214355139406377.
%Y A059933 Cf. A056193, A056004, A057650, A056041.
%Y A059933 Sequence in context: A116102 A013878 A058418 this_sequence A002488 A144692 
               A088469
%Y A059933 Adjacent sequences: A059930 A059931 A059932 this_sequence A059934 A059935 
               A059936
%K A059933 fini,nonn
%O A059933 2,1
%A A059933 Henry Bottomley (se16(AT)btinternet.com), Feb 12 2001
%E A059933 Definition corrected by N. J. A. Sloane (njas(AT)research.att.com), Mar 
               06 2006

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