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A056004 Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1. 20
0, 2, 3, 26, 27, 29, 30, 80, 81, 83, 84, 107, 108, 110, 111, 7625597484986, 7625597484987, 7625597484989, 7625597484990, 7625597485013, 7625597485014, 7625597485016, 7625597485017, 7625597485067, 7625597485068, 7625597485070 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

To write an integer n in base-k hereditary representation, write n in ordinary base-k representation, and then do the same recursively for all exponents which are greater than k: e.g., 2^18 = 2^(2^4 + 2) = 2^(2^(2^2) + 2). "Bump to base 3" means to replace all the 2's in that representation by 3. - M. F. Hasler, Feb 19 2017

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A. E. Caicedo, Goodstein's function, Revista Colombiana de Matemáticas 41 (2007), 381-391.

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

L. Kirby, and J. Paris, Accessible independence results for Peano arithmetic, Bull. London Mathematical Society, 14 (1982), 285-293.

Eric Weisstein's World of Mathematics, Heriditary Representation.

Eric Weisstein's World of Mathematics, Goodstein Sequence.

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

EXAMPLE

a(18)=7625597484989 since 18=2^(2^2)+2^1 which when bumped from 2 to 3 becomes 3^(3^3)+3^1=76255974849890 and when 1 is subtracted gives 7625597484989.

PROG

(Haskell)  see Link

(PARI) A056004(n)=sum(i=1, #n=binary(n), if(n[i], 3^if(#n-i<2, #n-i, A056004(#n-i)+1)))-1 \\ See A266201 for more general code. - M. F. Hasler, Feb 19 2017

CROSSREFS

Using G_k to denote the k-th step, this is the first in the following list: A056004: G_1(n), A057650: G_2(n), A059934: G_3(n), A059935: G_4(n), A059936: G_5(n); A266201: G_n(n); A056041.

Cf. A215409: G_n(3), A056193: G_n(4), A266204: G_n(5), A266205: G_n(6), A222117: G_n(15), A059933: G_n(16), A211378: G_n(19).

See A222112 for an alternate version.

Sequence in context: A060371 A130975 A002748 * A032812 A099006 A041659

Adjacent sequences:  A056001 A056002 A056003 * A056005 A056006 A056007

KEYWORD

nonn

AUTHOR

Henry Bottomley, Aug 04 2000

EXTENSIONS

Edited by M. F. Hasler, Feb 19 2017

STATUS

approved

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Last modified May 28 03:05 EDT 2017. Contains 287211 sequences.