A222113 -id:A222113 - cp's OEIS frontend

A266201 Goodstein numbers: a(n) = G_n(n), where G is the Goodstein function.

0, 0, 1, 2, 83, 1197, 187243, 37665879, 20000000211, 855935016215, 44580503598539, 2120126221988686, 155568095557812625, 6568408355712901455, 295147905179358418247, 14063084452070776884879

Offset: 0

Natan Arie Consigli, Jan 22 2016

nonn

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.
For example, the hereditary representation of 132132 in base-2 is:
132132 = 2^17 + 2^10 + 2^5 + 2^2
= 2^(2^4 + 1) + 2^(2^3 + 2) + 2^(2^2 + 1) + 2^2
= 2^(2^(2^2) + 1) + 2^(2^(2+1) + 2) + 2^(2^2 + 1) + 2^2.
Define B_k(n) to be the function that substitutes k+1 for all the bases of the base-k hereditary representation of n.
E.g., B_2(101) = B_2(2^(2^2 + 2) + 2^(2^2 + 1) + 2^2 + 1) = 3^(3^3 + 3) + 3^(3^3 + 1) + 3^3 + 1 = 228767924549638.
(Sometimes B_k(n) is referred to as n "bumped" from base k.)
The Goodstein function is defined as: G_k(n) = B_{k+1}(G_{k-1}(n)) - 1 with G_0(n) = n, i.e., iteration of bumping the number to the next larger base and subtracting one; see example section for instances.
Goodstein's theorem says that for any nonnegative n, the sequence G_k(n) eventually stabilizes and then decreases by 1 in each step until it reaches 0. (The subsequent values of G_k(n) < 0 are not part of the sequence.)
Named after the English mathematician Reuben Louis Goodstein (1912-1985). - Amiram Eldar, Jun 19 2021

			Compute a(5) = G_5(5):
G_0(5) = 5;
G_1(5) = B_2(G_0(5))-1 = B_2(2^2+1)-1 = (3^3+1)-1 = 27 = 3^3;
G_2(5) = B_3(G_1(5))-1 = B_3(3^3)-1 = 4^4-1 = 255 = 3*4^3+3*4^2+3*4+3;
G_3(5) = B_4(G_2(5))-1 = B_4(3*4^3+3*4^2+3*4+3)-1 = 467;
G_4(5) = B_5(G_3(5))-1 = B_5(3*5^3+3*5^2+3*5+2)-1 = 775;
G_5(5) = B_6(G_4(5))-1 = B_6(3*6^3+3*6^2+3*6+1)-1 = 1197.

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic, Vol. 9, No. 2 (1944), pp. 33-41; alternative link.
Eric Weisstein's World of Mathematics, Goodstein sequences.

Cf. Goodstein sequences: A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); 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).
Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A266202: g_n(n); A266203: a(n) = k such that g_k(n)=0;
Bumping Sequences: A222112: B_2(n);
Other sequences: A222113.

(B(n,b)=sum(i=1,#n=digits(n,b),n[i]*(b+1)^if(#nA266201(n)=for(k=1,n,n=B(n,k+1)-1);n \\ M. F. Hasler, Feb 12 2017

Edited by M. F. Hasler, Feb 12 2017

Incorrect a(16) deleted (the correct value is ~ 2.77*10^861) by M. F. Hasler, Feb 19 2017

A222117 Goodstein sequence starting with 15.

Original entry on oeis.org

15, 111, 1283, 18752, 326593, 6588344, 150994943, 3524450280, 100077777775, 3138578427934, 106993479003783, 3937376861542204, 155568096352467863, 6568408356994335930, 295147905181357143919, 14063084452070776884879, 708235345355342213988445

Offset: 0

Reinhard Zumkeller, Feb 13 2013

To calculate a(n+1), write a(n) in the hereditary representation base n+2, then bump the base to n+3, then subtract 1;

Compare to A222113: the underlying variants to define Goodstein sequences are equivalent.

			The first terms are:
a(0) = 2^(2+1) + 2^2 + 2^1 + 2^0 = 15;
a(1) = 3^(3+1) + 3^3 + 3^1 + 3^0 - 1 = 111;
a(2) = 4^(4+1) + 4^4 + 4^1 - 1 = 4^(4+1) + 4^4 + 3*4^0 = 1283;
a(3) = 5^(5+1) + 5^5 + 3*5^0 - 1 = 5^(5+1) + 5^5 + 2*5^0 = 18752;
a(4) = 6^(6+1) + 6^6 + 2*6^0 - 1 = 6^(6+1) + 6^6 + 1 = 326593;
a(5) = 7^(7+1) + 7^7 + 1 - 1 = 6588344;
a(6) = 8^(8+1) + 8^8 - 1 = 150994943.

Nicholas Matteo, Table of n, a(n) for n = 0..383 (first 249 terms from Reinhard Zumkeller)
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944
Eric Weisstein's World of Mathematics, Goodstein Sequence
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A215409 (start=3), A056193 (start=4), A059933 (start=16), A211378 (start=19).

Haskell
```
-- See Link
```

lista(nn) = {print1(a = 15, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); } \\ Michel Marcus, Feb 24 2016

Offset changed to 0 by Nicholas Matteo, Aug 21 2019

A265034 Weak Goodstein sequence beginning with 266.

Original entry on oeis.org

266, 6590, 65601, 390750, 1679831, 5765085, 16777579, 43047173, 100000551, 214359541, 429982475, 815731628, 1475790101, 2562891818, 4294968647, 6975758960, 11019962273, 16983564926, 25600002083, 37822861652, 54875876045, 78310988018, 110075317151, 152587893847

Offset: 0

N. J. A. Sloane, Dec 09 2015, following a suggestion from Alexander R. Povolotsky

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Antoine Nectoux, Goodstein Sequences: The Power of a Detour via Infinity, Klein Project Blog, 2015.

For other Goodstein sequences see A014221, A056004, A056041, A056193, A057650, A059933, A059934, A059935, A059936, A137411, A211378, A215409, A222112, A222113, A222117.

More terms from Chai Wah Wu, Dec 09 2015

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A266201 Goodstein numbers: a(n) = G_n(n), where G is the Goodstein function.

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A222117 Goodstein sequence starting with 15.

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A265034 Weak Goodstein sequence beginning with 266.

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