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.

16, 112, 1284, 18753, 326594, 6588345, 150994944, 3524450281, 100077777776, 3138578427935, 106993479003784, 3937376861542205, 155568096352467864, 6568408356994335931, 295147905181357143920, 14063084452070776884880, 708235345355342213988446

Offset

1,1

Comments

Compare to A222117: the underlying variants to define Goodstein sequences are equivalent.

References

Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

Example

a(1) - 1 = 15 = 2^3 + 2^2 + 2^1 + 2^0 = 2^(2^1+1) + 2^2 + 2^1 + 2^0
-> a(2) = 3^(3^1+1) + 3^3 + 3^1 + 3^0 = 112;
a(2) - 1 = 111 = 3^(3^1+1) + 3^3 + 3^1
-> a(3) = 4^(4^1+1) + 4^4 + 4^1 = 1284;
a(3) - 1 = 1283 = 4^(4^1+1) + 4^4 + 3*4^0
-> a(4) = 5^(5^1+1) + 5^5 + 3*5^0 = 18753;
a(4) - 1 = 18752 = 5^(5^1+1) + 5^5 + 2*5^0
-> a(5) = 6^(6^1+1) + 6^6 + 2*6^0 = 326594;
a(5) - 1 = 326593 = 6^(6^1+1) + 6^6 + 6^0
-> a(6)  = 7^(7^1+1) + 7^7 + 7^0 = 6588345.

Prog

(Haskell) -- See Link

Crossrefs

Cf. A222112.

Sequence in context: A000143 A258546 A205964 * A229531 A253460 A081194

Adjacent sequences: A222110 A222111 A222112 * A222114 A222115 A222116

Keyword

nonn,fini,changed

Author

Reinhard Zumkeller, Feb 13 2013

Status

approved