A059933 -id:A059933 - 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

A056193 Goodstein sequence starting with 4: 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

4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, 401, 458, 519, 584, 653, 726, 803, 884, 969, 1058, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2042, 2135, 2230, 2327, 2426, 2527, 2630, 2735, 2842, 2951, 3062, 3175, 3290, 3407

Offset: 0

Henry Bottomley, Aug 02 2000

Goodstein's theorem shows that such a sequence converges to zero for any starting value [e.g. if a(0)=1 then a(1)=0; if a(0)=2 then a(3)=0; and if a(0)=3 then a(5)=0]. With a(0)=4 we have a(3*2^(3*2^27 + 27) - 3)=0, which is well beyond the 10^(10^8)-th term.
The second half of such sequences is declining and the previous quarter is stable.
The resulting sequence 0,1,3,5,3*2^402653211 - 3, ... (see Comments in A056041) grows too rapidly to have its own entry.

			a(0) = 4 = 2^2,
a(1) = 3^3 - 1 = 26 = 2*3^2 + 2*3 + 2,
a(2) = 2*4^2 + 2*4 + 2 - 1 = 41 = 2*4^2 + 2*4 + 1,
a(3) = 2*5^2 + 2*5 + 1 - 1 = 60 = 2*5^2 + 2*5,
a(4) = 2*6^2 + 2*6 - 1 = 83 = 2*6^2 + 6 + 5,
a(5) = 2*7^2 + 7 + 5 - 1 = 109 etc.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (final 2 terms from Nicholas Matteo)
R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2 (1944), 33-41.
Eric Weisstein's World of Mathematics, Goodstein Sequence.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A056041, A056004, A057650, A059934, A059935, A059936, A271977.

Cf. A215409, A266204, A271554, A222117, A059933, A211378.

Haskell
```
See Zumkeller link
```

lista(nn) = {print1(a = 4, ", "); 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 22 2016

Edited by N. J. A. Sloane, Mar 06 2006

Offset changed to 0 by Nicholas Matteo, Sep 04 2019

A211378 Goodstein sequence starting with 19.

Original entry on oeis.org

19, 7625597484990, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084099

Offset: 0

Reinhard Zumkeller, Feb 13 2013

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

			The first terms are (see Wikipedia):
a(0) = 2^2^2 + 2^1 + 2^0 = 19
a(1) = 3^3^3 + 3^1 + 3^0 - 1 = 7625597484990
a(2) = 4^4^4 + 4^1 - 1                       (155 digits)
a(3) = 5^5^5 + 3 - 1                         (2185 digits)
a(4) = 6^6^6 + 2 - 1                         (36306 digits)
a(5) = 7^7^7 + 1 - 1                         (695975 digits)
a(6) = 8^8^8 - 1                             (15151336 digits).

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), A222117 (start=15), A059933 (start=16).

Haskell
```
-- See Link
```

Offset changed to 0 by Nicholas Matteo, Aug 21 2019

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

A215409 The Goodstein sequence G_n(3).

Original entry on oeis.org

3, 3, 3, 2, 1, 0

Offset: 0

Jonathan Sondow, Aug 10 2012

G_0(m) = m. To get the 2nd term, write m in hereditary base 2 notation (see links), change all the 2s to 3s, and then subtract 1 from the result. To get the 3rd term, write the 2nd term in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero (by Goodstein's Theorem), when the sequence terminates.

Decimal expansion of 33321/100000. - Natan Arie Consigli, Jan 23 2015

			a(0) = 3 = 2^1 + 1;
a(1) = 3^1 + 1 - 1 = 3^1 = 3;
a(2) = 4^1 - 1 = 3;
a(3) = 3 - 1 = 2;
a(4) = 2 - 1 = 1;
a(5) = 1 - 1 = 0.

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
Eric Weisstein's World of Mathematics, Hereditary Representation
Eric Weisstein's World of Mathematics, Goodstein Sequence
Eric Weisstein's World of Mathematics, Goodstein's Theorem
Wikipedia, Hereditary base-n notation
Wikipedia, Goodstein sequence
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences

Cf. A056004, A057650, A059934, A059935, A059936, A271977.

Cf. A056041, A056193, A266204, A271554, A222117, A059933, A211378.

Haskell
```
-- See Link
```

PadRight[CoefficientList[Series[3 + 3 x + 3 x^2 + 2 x^3 + x^4, {x, 0, 4}], x], 6] (* Michael De Vlieger, Dec 12 2017 *)

B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#nIain Fox, Dec 13 2017

first(n) = my(res = vector(n)); res[1] = res[2] = res[3] = 3; res[4] = 2; res[5] = 1; res; \\ Iain Fox, Dec 12 2017

first(n) = Vec(3 + 3*x + 3*x^2 + 2*x^3 + x^4 + O(x^n)) \\ Iain Fox, Dec 12 2017

a(n) = floor(2 - (4/Pi)*atan(n-3)) \\ Iain Fox, Dec 12 2017

a(0) = a(1) = a(2) = 3; a(3) = 2; a(4) = 1; a(n) = 0, n > 4;
From Iain Fox, Dec 12 2017: (Start)
G.f.: 3 + 3*x + 3*x^2 + 2*x^3 + x^4.
E.g.f.: 3 + 3*x + (3/2)*x^2 + (1/3)*x^3 + (1/24)*x^4.
a(n) = floor(2 - (4/Pi)*arctan(n-3)), n >= 0.
(End)

Corrected by Natan Arie Consigli, Jan 23 2015

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

A266205 a(n) = G_n(6), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, 1357259, 2011162, 2895965, 4068068, 5592391, 7542974, 10003577, 13068280, 16842083, 21441506, 26995189, 33644492, 41544095, 50862597, 61783119, 74503901, 89238903, 106218405, 125689607, 147917229

Offset: 0

Natan Arie Consigli, Jan 23 2016

			G_1(6) = B_2(6) - 1 = B_2(2^2 + 2) - 1 = 3^3 + 3 - 1 = 29;
G_2(6) = B_3(G_1(6)) - 1 = B_3(3^3 + 2) - 1 =  4^4 + 2 - 1 = 257;
G_3(6) = B_4(G_2(6)) - 1 = 5^5 + 1 - 1 = 3125;
G_4(6) = B_5(G_3(6)) - 1 = 6^6 - 1 = 46655;
G_5(6) = B_6(G_4(6)) - 1 = 5*7^5 + 5*7^4 + 5*7^3 + 5*7^2 + 5*7 + 5 - 1 = 98039.

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).

lista(nn) = {print1(a = 6, ", "); 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 22 2016

A059936 Fifth step in Goodstein sequences, i.e., g(7) if g(2)=n: write g(6)=A059935(n) in hereditary representation base 6, bump to base 7, then subtract 1 to produce g(7).

Original entry on oeis.org

0, 109, 1197, 98039, 823543, 1647195, 2471826, 4215754, 5764801, 5764910, 5765998, 5862840, 6588344, 5103708485122940631839901111036829791435007685667303872450435153015345686896530517814322070729709

Offset: 3

Henry Bottomley, Feb 12 2001

nonn

a(17) = 4.587...*10^1014, a(18) = 1.505...*10^82854, and 3.759...*10^695974 = a(19) < a(20) < ... < a(31) = a(19) + 6588344. - Pontus von Brömssen, Sep 20 2020

			a(12) = 5764910 since with g(2) = 12 = 2^(2 + 1) + 2^2, we get g(3) = 3^(3 + 1) + 3^3-1 = 107 = 3^(3 + 1) + 2*3^2 + 2*3 + 2, g(4) = 4^(4 + 1) + 2*4^2 + 2*4 + 1 = 1065, g(5) = 5^(5 + 1) + 2*5^2 + 2*5 = 15685, g(6) = 6^(6 + 1) + 2*6^2 + 6 + 5 = 280019 and g(7) = 7^(7 + 1) + 2*7^2 + 7 + 4 = 5764910.

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. A056004, A057650, A059933, A059934, A059935.

Haskell
```
-- See Link
```

from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A059936(n):
  for i in range(2,7):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 19 2020

a(16) corrected by Pontus von Brömssen, Sep 18 2020

A057650 Second step in Goodstein sequences, i.e., g(4) if g(2)=n: (first step) write g(2)=n in hereditary representation base 2, bump to base 3, then subtract 1 to produce g(3)=A056004(n), then (second step) write g(3) in hereditary representation base 3, bump to base 4, then subtract 1 to produce g(4).

Original entry on oeis.org

1, 3, 41, 255, 257, 259, 553, 1023, 1025, 1027, 1065, 1279, 1281, 1283, 50973998591214355139406377, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084095

Offset: 2

Henry Bottomley, Oct 13 2000

nonn

			a(12)=1065 since with g(2) = 12 = 2^(2+1) + 2^2, we get g(3) = 3^(3+1) + 3^3 - 1 = 107 = 3^(3+1) + 2*3^2 + 2*3 + 2 and g(4) = 4^(4+1) + 2*4^2 + 2*4 + 2 - 1 = 1065.
a(17) = 4^(4^4) - 1, with g(2) = 17 = 2^(2^2) + 1 and g(3) = 3^(3^3).
Similarly a(18) = 4^(4^4) + 1, with g(2) = 18 = 2^(2^2) + 2 and g(3) = 3^(3^3) + 2.

Reinhard Zumkeller, Table of n, a(n) for n = 2..1000
R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
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

Cf. A056004, A059933, A059934, A059935, A059936.

Haskell
```
-- See Link
```

cp's OEIS Frontend

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

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A056193 Goodstein sequence starting with 4: 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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A211378 Goodstein sequence starting with 19.

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A222117 Goodstein sequence starting with 15.

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A215409 The Goodstein sequence G_n(3).

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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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