login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS Foundation is grateful to everyone who made a donation during our Annual Appeal.     Visit the new and spectacular Pictures from the OEIS page!

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056193 Goodstein sequence with a(0)=4: to calculate a(n+1), write a(n) in the hereditary representation base n, then bump the base to n+1, then subtract 1. 8
4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, 401, 458, 519, 584, 653, 726, 803, 884, 969, 1058, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2042, 2135, 2230, 2327, 2426, 2527, 2630, 2735, 2842, 2951, 3062, 3175, 3290, 3407 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Goodstein's theorem shows that such sequence converges to zero for any starting value [e.g. if a(2)=1 then a(3)=0; if a(2)=2 then a(5)=0; and if a(2)=3 then a(7)=0]. With a(2)=4 we have a(3*2^(3*2^27+27)-1)=0, which is well beyond the 10^(10^8)-th term.

The second half of such sequences is declining and the previous quarter is stable.

The resulting sequence 2,3,5,7,3*2^402653211 - 1, ... (see Comments in A056041) grows too rapidly to have its own entry.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.

Eric Weisstein's World of Mathematics, Goodstein Sequence.

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

EXAMPLE

a(2)=4=2^2, a(3)=3^3-1=26=2*3^2+2*3+2, a(4)=2*4^2+2*4+2-1=41=2*4^2+2*4+1, a(5)=2*5^2+2*5+1-1=60=2*5^2+2*5, a(6)=2*6^2+2*6-1=83=2*6^2+6+5, a(7)=2*7^2+7+5-1=109 etc.

PROG

(Haskell)  see Link

CROSSREFS

Cf. A056041, A056004, A059934, A057650, A059933, A059935, A059936.

Cf. A215409, A222117, A211378.

Sequence in context: A046963 A022386 A059178 * A196672 A102203 A219668

Adjacent sequences:  A056190 A056191 A056192 * A056194 A056195 A056196

KEYWORD

nonn

AUTHOR

Henry Bottomley, Aug 02 2000

EXTENSIONS

Edited by N. J. A. Sloane, Mar 06 2006

Corrected by Natan Arie' Consigli Jan 23 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 13 01:52 EST 2016. Contains 268229 sequences.