A059934 -id:A059934 - cp's OEIS frontend

A059935 Fourth step in Goodstein sequences, i.e., g(6) if g(2)=n: write g(5)=A059934(n) in hereditary representation base 5, bump to base 6, then subtract 1 to produce g(6).

1, 83, 775, 46655, 46657, 93395, 140743, 279935, 279937, 280019, 280711, 326591, 326593, 19916489515870532960258562190639398471599239042185934648024761145811

Offset: 3

Henry Bottomley, Feb 12 2001

nonn

2.659...*10^36305 = a(18) < a(19) < ... < a(31) = a(18) + 326594. - Pontus von Brömssen, Sep 20 2020

			a(12) = 280019 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 and g(6) = 6^(6 + 1) + 2*6^2 + 6 + 5 = 280019.

Pontus von Brömssen, Table of n, a(n) for n = 3..17
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, A059936.

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 A059935(n):
  for i in range(2,6):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 20 2020

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

Original entry on oeis.org

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

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

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

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
```

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

A271977 G_6(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 139, 1751, 187243, 16777215, 33554571, 50333399, 84073323, 134217727, 134217867, 134219479, 134404971, 150994943

Offset: 3

Natan Arie Consigli, Apr 24 2016

nonn

The next term (line break for better formatting) is a(16) = \
1619239197880733074062994004113160848331305687934176134326809 \
538279709713884753268291640071900343455846003089194770060104834018705547.
a(17) = 2.870...*10^1585, a(18) = 6.943...*10^169099. - Pontus von Brömssen, Sep 24 2020

			Find G_6(7):
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.

Pontus von Brömssen, Table of n, a(n) for n = 3..16
Wikipedia, Goodstein's theorem

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); this sequence: G_6(n); A271978: G_7(n); A271979: G_8(n); A271985: G_9(n); A271986: G_10(n); A266201: G_n(n).

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 A271977(n):
  if n==3: return 0
  for i in range(2,8):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 24 2020

a(10) corrected by Pontus von Brömssen, Sep 24 2020

A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

Original entry on oeis.org

2, 3, 5, 7, 23, 63, 383, 2047

Offset: 0

Henry Bottomley, Aug 04 2000

a(8)=3*2^(3*2^27+27)-1 which is more than 10^(10^8) and equal to the final base of the Goodstein sequence starting with g(2)=4; indeed, apart from the initial term, the sequence starting with b(2)=8 is identical to the Goodstein sequence starting with g(2)=4. The initial terms of a(n) [2, 3, 5 and 7] are equal to the initial terms of the equivalent final bases of Goodstein sequences starting at the same points. a(9)=2^(2^(2^70+70)+2^70+70)-1 which is more than 10^(10^(10^20)).
It appears that if n is even then a(n) is one less than three times a power of two, while if n is odd then a(n) is one less than a power of two.
Comment from John Tromp, Dec 02 2004: The sequence 2,3,5,7,3*2^402653211 - 1, ... gives the final base of the Goodstein sequence starting with n. This is an example of a very rapidly growing function that is total (i.e. defined on any input), although this fact is not provable in first-order Peano Arithmetic. See the links for definitions. This grows even faster than the Friedman sequence described in the Comments to A014221.
In fact there are two related sequences: (i) The Goodstein function l(n) = number of steps for the Goodstein sequence to reach 0 when started with initial term n >= 0: 0, 1, 3, 5, 3*2^402653211 - 3, ...; and (ii) the same sequence + 2: 2, 3, 5, 7, 3*2^402653211 - 1, ..., which is the final base reached. Both grow too rapidly to have their own entries in the database.
Related to the hereditary base sequences - see cross-reference lines.
This sequence gives the final base of the weak Goodstein sequence starting with n; compare A266203, the length of the weak Goodstein sequence. a(n) = A266203(n) + 2.

			a(3)=7 because starting with b(2)=3=11 base 2, we get b(3)=11-1 base 3=10 base 3=3, b(4)=10-1 base 4=3, b(5)=3-1 base 5=2, b(6)=2-1 base 6=1 and b(7)=1-1 base 7=0.

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.
J. Tromp, Programming Pearls
Eric Weisstein's World of Mathematics, Goodstein Sequence
Wikipedia, Goodstein's theorem

Cf. A266202, A268687, A268689, A268688.
Equals A266203 + 2.
Weak Goodstein sequences: A267647, A267648, A271987, A271988, A271989, A271990, A271991, A137411, A271992, A265034.
Steps of strong Goodstein sequences: A056004, A057650, A059934, A059935, A059936, A271977.
Strong Goodstein sequences: A215409, A056193, A266204, A222117, A059933.
Woodall numbers: A003261.

cp's OEIS Frontend

A059935 Fourth step in Goodstein sequences, i.e., g(6) if g(2)=n: write g(5)=A059934(n) in hereditary representation base 5, bump to base 6, then subtract 1 to produce g(6).

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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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A215409 The Goodstein sequence G_n(3).

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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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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).

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