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A056004
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Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1.
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21
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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, Hereditary Representation.
Eric Weisstein's World of Mathematics, Goodstein Sequence.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, Aug 04 2000
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EXTENSIONS
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Edited by M. F. Hasler, Feb 19 2017
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STATUS
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approved
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