A056004 Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1.
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, 7625597485071, 7625597485094
Offset: 1
Keywords
Examples
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.
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, Hereditary Representation.
- Eric Weisstein's World of Mathematics, Goodstein Sequence.
- Wikipedia, Goodstein's Theorem
- Reinhard Zumkeller, Haskell programs for Goodstein sequences
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.
Programs
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Haskell
-- See Link
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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
Extensions
Edited by M. F. Hasler, Feb 19 2017
Comments