A266205 -id:A266205 - 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

A059933 Goodstein sequence starting with 16: to calculate a(n+1), write a(n) in the hereditary representation in base n+2, then bump the base to n+3, then subtract 1.

Original entry on oeis.org

16, 7625597484986, 50973998591214355139406377, 53793641718868912174424175024032593379100060, 19916489515870532960258562190639398471599239042185934648024761145811, 5103708485122940631839901111036829791435007685667303872450435153015345686896530517814322070729709

Offset: 0

Henry Bottomley, Feb 12 2001

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

			a(0) = 16 = 2^(2^2) so a(1) = 3^(3^3)-1 = 7625597484986.
So a(1) = 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(2) = 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.

Reinhard Zumkeller, Table of n, a(n) for n = 0..18
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

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).

Haskell
```
-- See Link
```

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

(B(n,b)=sum(i=1,#n=digits(n,b),n[i]*(b+1)^if(#n1,B(a,n)-1,16)) \\ M. F. Hasler, Feb 12 2017

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

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
Offset changed to 0 by Nicholas Matteo, Aug 21 2019

A266204 a(n) = G_n(5), where G_n(k) is the Goodstein function defined in A266201.

Original entry on oeis.org

5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382, 5643, 7126, 8849, 10830, 13087, 15637, 18499, 21691, 25231, 29137, 33427, 38119, 43231, 48781, 54787, 61267, 68239, 75721, 83731, 92287, 101407, 111108, 121409, 132328, 143883, 156092, 168973, 182544, 196823

Offset: 0

Natan Arie Consigli, Jan 22 2016

			G_0(5) = 5;
G_1(5) = B_2(5) - 1 = B_2(2^2 + 1) - 1 = 27;
G_2(5) = B_3(3^3) - 1 = 4^4 - 1 = 255;
G_3(5) = B_4(3*4^3 + 3*4^2 + 3*4 + 3) - 1 = 3*5^3 + 3*5^2 + 3*5 + 3 - 1 = 467.

Nicholas Matteo, Table of n, a(n) for n = 0..10000
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
Wikipedia, Goodstein sequence

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A059936: G_5(n), A266201: G_n(n).

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 = 5, ", "); for (n=2, nn, a = bump(a, n)-1; print1(a, ", "); ); } \\ Michel Marcus, Feb 28 2016

A056004 Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1.

Original entry on oeis.org

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

Henry Bottomley, Aug 04 2000

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: 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

			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.

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

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.

Haskell
```
-- See Link
```

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

Edited by M. F. Hasler, Feb 19 2017

A271554 a(n) = G_n(7), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

7, 30, 259, 3127, 46657, 823543, 16777215, 37665879, 77777775, 150051213, 273624711, 475842915, 794655639, 1281445305, 2004318063, 3051893870, 4537630813, 6604718946, 9431578931, 13238000758, 18291957825, 24917131658, 33501182551, 44504801406, 58471578053, 76038721330

Offset: 0

Natan Arie Consigli, Apr 10 2016

			G_1(7) = B_2(7) - 1 = B[2](2^2 + 2 + 1) - 1 = 3^3 + 3 + 1 - 1 = 30;
G_2(7) = B_3(G_1(7)) - 1 = B[3](3^3 + 3) - 1 =  4^4 + 4 - 1 = 259;
G_3(7) = B_4(G_2(7)) - 1 = 5^5 + 3 - 1 = 3127;
G_4(7) = B_5(G_3(7)) - 1 = 6^6 + 2 - 1 = 46657;
G_5(7) = B_6(G_4(7)) - 1 = 7^7 + 1 - 1 = 823543;
G_6(7) = B_7(G_5(7)) - 1 = 8^8 - 1 = 16777215;
G_7(7) = B_8(G_6(7)) - 1 = 7*9^7 + 7*9^6 + 7*9^5 + 7*9^4 + 7*9^3 + 7*9^2 + 7*9 + 7 - 1 = 37665879.

Nicholas Matteo, Table of n, a(n) for n = 0..10000
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
Wikipedia, Goodstein sequence

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A266201: G_n(n).

lista(nn) = {print1(a = 7, ", "); 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, ", "); ); }

A271555 a(n) = G_n(8), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

8, 80, 553, 6310, 93395, 1647195, 33554571, 774841151, 20000000211, 570623341475, 17832200896811, 605750213184854, 22224013651116433, 875787780761719208, 36893488147419103751, 1654480523772673528938, 78692816150593075151501, 3956839311320627178248684

Offset: 0

Natan Arie Consigli, Apr 10 2016

			G_1(8) = B_2(8)-1 = B_2(2^(2+1))-1 = 3^(3+1)-1 = 80;
G_2(8) = B_3(2*3^3+2*3^2+2*3+2)-1 = 2*4^4+2*4^2+2*4+2-1 = 553;
G_3(8) = B_4(2*4^4+2*4^2+2*4+1)-1 = 2*5^5+2*5^2+2*5+1-1 = 6310;
G_4(8) = B_5(2*5^5+2*5^2+2*5)-1 = 2*6^6+2*6^2+2*6-1 = 93395;
G_5(8) = B_6(2*6^6+2*6^2+6+5)-1 = 2*7^7+2*7^2+7+5-1 = 1647195;
G_6(8) = B_7(2*7^7+2*7^2+7+4)-1 = 2*8^8+2*8^2+8+4-1 = 33554571;
G_7(8) = B_8(2*8^8+2*8^2+8+3)-1 = 2*9^9+2*9^2+9+3-1 = 774841151.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
Wikipedia, Goodstein sequence

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A266201: G_n(n).

lista(nn) = {print1(a = 8, ", "); 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, ", "); ); }

a(3) corrected by Nicholas Matteo, Aug 15 2019

A271556 a(n) = G_n(9), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

9, 81, 1023, 9842, 140743, 2471826, 50333399, 1162263921, 30000003325, 855935016215, 26748301350411, 908625319783885, 33336020476682897, 1313681671142588955, 55340232221128667935, 2481720785659010308168, 118039224225889612744771, 5935258966980940767393628

Offset: 0

Natan Arie Consigli, Apr 10 2016

			G_1(9) = B_2(9)-1 = B_2(2^(2+1)+1)-1 = 3^(3+1) + 1-1 = 81;
G_2(9) = B_3(3^(3+1))-1 = 4^(4+1)-1 = 1023;
G_3(9) = B_4(3*4^4 + 3*4^3 + 3*4^2 + 3*4 + 3)-1 = 3*5^5 + 3*5^3 + 3*5^2 + 3*5 + 3-1 = 9842;
G_4(9) = B_5(3*5^5 + 3*5^3 + 3*5^2 + 3*5 + 2)-1 = 3*6^6 + 3*6^3 + 3*6^2 + 3*6 + 2-1 = 140743;
G_5(9) = B_6(3*6^6 + 3*6^3 + 3*6^2 + 3*6 + 1)-1 = 3*7^7 + 3*7^3 + 3*7^2 + 3*7 + 1-1 = 2471826;
G_6(9) = B_7(3*7^7 + 3*7^3 + 3*7^2 + 3*7)-1 = 3*8^8 + 3*8^3 + 3*8^2 + 3*8-1 = 50333399.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A266201: G_n(n).

lista(nn) = {print1(a = 9, ", "); 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, ", "); ); }

A271557 a(n) = G_n(10), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

10, 83, 1025, 15625, 279935, 4215754, 84073323, 1937434592, 50000555551, 1426559238830, 44580503598539, 1514375534972427, 55560034130686045, 2189469451908364943, 92233720368553350471, 4136201309431691363859, 196732040376482697880697, 9892098278301567958688175

Offset: 0

Natan Arie Consigli, Apr 11 2016

			G_1(10) = B_2(10)-1 = B_2(2^(2+1)+2)-1 = 3^(3+1)+3-1 = 83;
G_2(10) = B_3(3^(3+1)+2)-1 = 4^(4+1)+2-1 = 1025;
G_3(10) = B_4(4^(4+1)+1)-1 = 5^(5+1)+1-1 = 15625;
G_4(10) = B_5(5*5^(5+1))-1 = 6^(6+1)-1= 279935;
G_5(10) = B_6(5*6^6+5*6^5+5*6^4+5*6^3+5*6^2+5*6+5)-1 = 5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+5-1 = 4215754;
G_6(10) = B_7(5*7^7+5*7^5+5*7^4+5*7^3+5*7^2+5*7+4)-1 = 5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+4-1 = 84073323;
G_7(10) = B_8(5*8^8+5*8^5+5*8^4+5*8^3+5*8^2+5*8+3)-1 = 5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+3-1 = 1937434592;
G_8(10) = B_9(5*9^9+5*9^5+5*9^4+5*9^3+5*9^2+5*9+2)-1 = 5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+2-1 = 50000555551.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A266201: G_n(n).

lista(nn) = {print1(a = 10, ", "); 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, ", "); ); }

A222112 Initial step in Goodstein sequences: write n-1 in hereditary binary representation, then bump to base 3.

Original entry on oeis.org

0, 1, 3, 4, 27, 28, 30, 31, 81, 82, 84, 85, 108, 109, 111, 112, 7625597484987, 7625597484988, 7625597484990, 7625597484991, 7625597485014, 7625597485015, 7625597485017, 7625597485018, 7625597485068, 7625597485069, 7625597485071, 7625597485072, 7625597485095

Offset: 1

Reinhard Zumkeller, Feb 13 2013

nonn

See A056004 for an alternate version.

			n = 19: 19 - 1 = 18 = 2^4 + 2^1 = 2^2^2 + 2^1
-> a(19) = 3^3^3 + 3^1 = 7625597484990;
n = 20: 20 - 1 = 19 = 2^4 + 2^1 + 2^0 = 2^2^2 + 2^1 + 2^0
-> a(20) = 3^3^3 + 3^1 + 3^0 = 7625597484991;
n = 21: 21 - 1 = 20 = 2^4 + 2^2 = 2^2^2 + 2^2
-> a(21) = 3^3^3 + 3^3 = 7625597485014.

Helmut Schwichtenberg and Stanley S. Wainer, Proofs and Computations, Cambridge University Press, 2012; 4.4.1, page 148ff.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A056004: G_1(n), A057650 G_2(n), A056041; A266201: G_n(n);

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).

Haskell
```
-- See Link
```

A222112(n)=sum(i=1, #n=binary(n-1), if(n[i],3^if(#n-i<2, #n-i, A222112(#n-i+1)))) \\ See A266201 for more general code. - M. F. Hasler, Feb 13 2017, edited Feb 19 2017

A271558 a(n) = G_n(11), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

11, 84, 1027, 15627, 279937, 5764801, 134217727, 2749609302, 70077777775, 1997331745490, 62412976762503, 2120126221988686, 77784048573561751, 3065257233947460930, 129127208517971179375, 5790681833207409243109, 275424856527080300658781, 13848937589622201728586799

Offset: 0

Natan Arie Consigli, Apr 11 2016

			G_1(11) = B_2(11)-1 = B_2(2^(2+1)+2+1)-1 = 3^(3+1)+3+1-1 = 84;
G_2(11) = B_3(3^(3+1)+3)-1 = 4^(4+1)+4-1 = 1027;
G_3(11) = B_4(4^(4+1)+3)-1 = 5^(5+1)+3-1 = 15627;
G_4(11) = B_5(5^(5+1)+2)-1 = 6^(6+1)+2-1 = 279937;
G_5(11) = B_6(6^(6+1)+1)-1 = 7^(7+1)+1-1 = 5764801;
G_6(11) = B_7(7^(7+1))-1 = 8^(8+1)-1 = 134217727.

Nicholas Matteo, Table of n, a(n) for n = 0..384
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.

Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A266201: G_n(n).

lista(nn) = {print1(a = 11, ", "); 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, ", "); ); }

a(9)-a(13) corrected by Nicholas Matteo, Aug 15 2019

a(14) onwards from Nicholas Matteo, Aug 28 2019

cp's OEIS Frontend

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

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A059933 Goodstein sequence starting with 16: to calculate a(n+1), write a(n) in the hereditary representation in base n+2, then bump the base to n+3, then subtract 1.

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A266204 a(n) = G_n(5), where G_n(k) is the Goodstein function defined in A266201.

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A056004 Initial step in Goodstein sequences: write n in hereditary representation base 2, bump to base 3, then subtract 1.

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A271554 a(n) = G_n(7), where G is the Goodstein function defined in A266201.

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A271555 a(n) = G_n(8), where G is the Goodstein function defined in A266201.

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A271556 a(n) = G_n(9), where G is the Goodstein function defined in A266201.

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A271557 a(n) = G_n(10), where G is the Goodstein function defined in A266201.

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