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

Author

Reinhard Zumkeller, Feb 13 2013

Keywords

• nonn
• fini

Comments

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

Examples

			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.
		

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

Crossrefs

Cf. A222112.

Programs

Haskell

-- See Link