%I #38 Apr 24 2024 11:21:10

%S 0,2,3,26,27,29,30,80,81,83,84,107,108,110,111,7625597484986,

%T 7625597484987,7625597484989,7625597484990,7625597485013,

%U 7625597485014,7625597485016,7625597485017,7625597485067,7625597485068,7625597485070,7625597485071,7625597485094

%N Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1.

%C 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

%H Reinhard Zumkeller, <a href="/A056004/b056004.txt">Table of n, a(n) for n = 1..10000</a>

%H A. E. Caicedo, <a href="http://www.scielo.org.co/pdf/rcm/v41n2/v41n2a08.pdf">Goodstein's function</a>, Revista Colombiana de Matemáticas 41 (2007), 381-391.

%H R. L. Goodstein, <a href="http://www.jstor.org/stable/2268019">On the Restricted Ordinal Theorem</a>, J. Symb. Logic 9, 33-41, 1944.

%H L. Kirby, and J. Paris, <a href="http://blms.oxfordjournals.org/content/14/4/285.extract">Accessible independence results for Peano arithmetic</a>, Bull. London Mathematical Society, 14 (1982), 285-293.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HereditaryRepresentation.html">Hereditary Representation.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>

%H Reinhard Zumkeller, <a href="/A211378/a211378.hs.txt">Haskell programs for Goodstein sequences</a>

%e 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.

%o (Haskell) -- See Link

%o (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

%Y 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.

%Y 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).

%Y See A222112 for an alternate version.

%K nonn

%O 1,2

%A _Henry Bottomley_, Aug 04 2000

%E Edited by _M. F. Hasler_, Feb 19 2017