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

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056193 Goodstein sequence with a(2)=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. 7
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

2,1

COMMENTS

Goodstein's theorem shows that such a sequence is finite (i.e. eventually reaches 0) 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.

REFERENCES

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

LINKS

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

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

fini,nonn

AUTHOR

Henry Bottomley, Aug 02 2000

EXTENSIONS

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

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.

Content is available under The OEIS End-User License Agreement .

Last modified December 20 23:03 EST 2014. Contains 252290 sequences.