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A059933 Goodstein sequence with a(2)=16: to calculate a(n+1), write a(n) in the hereditary representation in base n, then bump the base to n+1, then subtract 1. 30
16, 7625597484986, 50973998591214355139406377, 53793641718868912174424175024032593379100060, 19916489515870532960258562190639398471599239042185934648024761145811, 5103708485122940631839901111036829791435007685667303872450435153015345686896530517814322070729709 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

See A266201 for definitions of and key links for hereditary representation and Goodstein sequences.

Goodstein's theorem shows that the Goodstein sequence G_n(k) eventually stabilizes and then decreases by 1 at each step until it reaches 0. Thereafter the values of G_n(k) < 0 are not part of the sequence. By Goodstein's theorem we conclude that G_n(k) is a finite sequence.

In this case when a(2) = G_0(16) = 16, there seems little possibility of describing how incredibly large n must be for a(n) to reach 0.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..20

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

Eric Weisstein's World of Mathematics, Goodstein Sequence

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

FORMULA

a(n+2) = G_n(16), where G is the function defined in A266201.

EXAMPLE

a(2) = 16 = 2^(2^2) so a(3) = 3^(3^3)-1 = 7625597484986.

So a(3) = 2*3^(2*3^2 + 2*3 + 2) + 2*3^(2*3^2 + 2*3 + 1) + 2*3^(2*3^2 + 2*3) + 2*3^(2*3^2 + 1*3 + 2) + 2*3^(2*3^2 + 1*3 + 1) + 2*3^(2*3^2 + 1*3) + 2*3^(2*3^2 + 2) + 2*3^(2*3^2 + 1) + 2*3^(2*3^2) + 2*3^(3^2 + 2*3 + 2) + 2*3^(3^2 + 2*3 + 1) + 2*3^(3^2 + 2*3) + 2*3^(3^2 + 1*3 + 2) + 2*3^(3^2 + 1*3 + 1) + 2*3^(3^2 + 1*3) + 2*3^(3^2 + 2) + 2*3^(3^2 + 1) + 2*3^(3^2) + 2*3^(2*3 + 2) + 2*3^(2*3 + 1) + 2*3^(2*3) + 2*3^(1*3 + 2) + 2*3^(1*3 + 1) + 2*3^(1*3) + 2*3^(2) + 2*3^(1) + 2,

leading to a(4) = 2*4^(2*4^2 + 2*4 + 2) + 2*4^(2*4^2 + 2*4 + 1) + 2*4^(2*4^2 + 2*4) + 2*4^(2*4^2 + 1*4 + 2) + 2*4^(2*4^2 + 1*4 + 1) + 2*4^(2*4^2 + 1*4) + 2*4^(2*4^2 + 2) + 2*4^(2*4^2 + 1) + 2*4^(2*4^2) + 2*4^(4^2 + 2*4 + 2) + 2*4^(4^2 + 2*4 + 1) + 2*4^(4^2 + 2*4) + 2*4^(4^2 + 1*4 + 2) + 2*4^(4^2 + 1*4 + 1) + 2*4^(4^2 + 1*4) + 2*4^(4^2 + 2) + 2*4^(4^2 + 1) + 2*4^(4^2) + 2*4^(2*4 + 2) + 2*4^(2*4 + 1) + 2*4^(2*4) + 2*4^(1*4 + 2) + 2*4^(1*4 + 1) + 2*4^(1*4) + 2*4^(2) + 2*4^(1) + 1 = 2*(4^32 + 4^16 + 1)*(4^8 + 4^4 + 1)*(4^2 + 4*1)-1 = 50973998591214355139406377.

PROG

(Haskell)  see Link

(PARI) bump(a, n) = {if (a < n, return (a)); my(pd = Pol(digits(a, n)));  my(de = vector(poldegree(pd)+1, k, k--; polcoeff(pd, k))); my(bde = vector(#de, k, k--; bump(k, n))); my(q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^bde[k+1], 0))); return(subst(q, x, n+1)); }

lista(nn) = {print1(a = 16, ", "); for (n=2, nn, a = bump(a, n)-1; print1(a, ", "); ); } \\ Michel Marcus, Feb 28 2016

(PARI) (B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n<b+i, #n-i, B(#n-i, b)))); vector(7, n, a=if(n>1, B(a, n)-1, 16)) \\ M. F. Hasler, Feb 12 2017

CROSSREFS

Cf. A266201: G_n(n).

Cf. A056193: G_n(4), A056004: G_1(n), A057650 G_2(n), A056041.

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

Sequence in context: A013878 A058418 A291908 * A002488 A243776 A198631

Adjacent sequences:  A059930 A059931 A059932 * A059934 A059935 A059936

KEYWORD

nonn,hard,fini

AUTHOR

Henry Bottomley, Feb 12 2001

EXTENSIONS

Definition corrected by N. J. A. Sloane, Mar 06 2006

Missing a(5) inserted and wrong a(7) replaced by Reinhard Zumkeller, Feb 13 2013

Revised by Natan Arie' Consigli, Jan 23 2016

STATUS

approved

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Last modified July 24 05:48 EDT 2019. Contains 325290 sequences. (Running on oeis4.)