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%I A222113 #12 Apr 24 2024 11:13:56
%S A222113 16,112,1284,18753,326594,6588345,150994944,3524450281,100077777776,
%T A222113 3138578427935,106993479003784,3937376861542205,155568096352467864,
%U A222113 6568408356994335931,295147905181357143920,14063084452070776884880,708235345355342213988446
%N A222113 Goodstein sequence starting with a(1) = 16: to calculate a(n) for n>1, subtract 1 from a(n-1) and write the result in the hereditary representation base n, then bump the base to n+1.
%C A222113 Compare to A222117: the underlying variants to define Goodstein sequences are equivalent.
%D A222113 Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.
%H A222113 Reinhard Zumkeller, Table of n, a(n) for n = 1..250
%H A222113 R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.
%H A222113 Wikipedia, Goodstein's Theorem
%H A222113 Reinhard Zumkeller, Haskell programs for Goodstein sequences
%e A222113 a(1) - 1 = 15 = 2^3 + 2^2 + 2^1 + 2^0 = 2^(2^1+1) + 2^2 + 2^1 + 2^0
%e A222113 -> a(2) = 3^(3^1+1) + 3^3 + 3^1 + 3^0 = 112;
%e A222113 a(2) - 1 = 111 = 3^(3^1+1) + 3^3 + 3^1
%e A222113 -> a(3) = 4^(4^1+1) + 4^4 + 4^1 = 1284;
%e A222113 a(3) - 1 = 1283 = 4^(4^1+1) + 4^4 + 3*4^0
%e A222113 -> a(4) = 5^(5^1+1) + 5^5 + 3*5^0 = 18753;
%e A222113 a(4) - 1 = 18752 = 5^(5^1+1) + 5^5 + 2*5^0
%e A222113 -> a(5) = 6^(6^1+1) + 6^6 + 2*6^0 = 326594;
%e A222113 a(5) - 1 = 326593 = 6^(6^1+1) + 6^6 + 6^0
%e A222113 -> a(6) = 7^(7^1+1) + 7^7 + 7^0 = 6588345.
%o A222113 (Haskell) -- See Link
%Y A222113 Cf. A222112.
%K A222113 nonn,fini
%O A222113 1,1
%A A222113 _Reinhard Zumkeller_, Feb 13 2013
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