A266201 -id:A266201 - cp's OEIS frontend

A271978 G_7(n), where G is the Goodstein function defined in A266201.

0, 173, 2454, 332147, 37665879, 774841151, 1162263921, 1937434592, 2749609302, 3486784574, 3486786855, 3487116548, 3524450280

Offset: 3

Natan Arie Consigli, Apr 30 2016

nonn

a(16) is too big to include - see b-file. a(17) = 9.221...*10^2347, a(18) = 2.509...*10^316952. - Pontus von Brömssen, Sep 25 2020

			Find G_7(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;
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.

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); A271977: G_6(n); this sequence: 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 A271978(n):
  if n==3: return 0
  for i in range(2,9):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

a(9) corrected by Pontus von Brömssen, Sep 25 2020

A271979 G_8(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 211, 3325, 555551, 77777775, 20000000211, 30000003325, 50000555551, 70077777775, 100000000211, 100000003325, 100000555551, 100077777775

Offset: 3

Natan Arie Consigli, Apr 30 2016

nonn

At least half of the digits of every term (except a(14)) are the same.
Let n > 0:
a(4n) mod 100 = 211;
a(4n+1) mod 1000 = 3325;
a(4n+2) mod 1000000 = 555551;
a(4n+3) mod 100000000 = 77777775;
Proof for a(4n):
If x is divisible by 4 its hereditary representation in base 2 has all summands divisible by 4 and it cannot have the summands 1 and 2.
If we calculate G_1(x) we would end with:
G_1(x) = B_2(x)-1.
Clearly, B_2(x) = 3^a + 3^b + ... is divisible by 3^3 = 27 and that would mean that the representation of B_2(x)-1 would be B_2(x)-1 = X_3 + 2*3^2+2*3+2.
From now on, let X_n be a sum of powers of n (greater than the right term).
We finish proving the statement by calculating G_8(x):
G_2(x) = B_3(X_3 +2*3^2+2*3+2)-1 = X_4 + 2*4^2+2*4+2-1;
G_3(x) = B_4(X_4 +2*4^2+2*4-1)-1 = X_5 + 2*5^2+2*5+1-1;
G_4(x) = B_5(X_5 +2*5^2+2*5)-1 = X_6 + 2*6^2+2*6-1;
G_5(x) = B_6(X_6 +2*6^2+6+5)-1 = X_7 + 2*7^2+7+5-1;
G_6(x) = B_7(X_7 +2*7^2+7+4)-1 = X_8 + 2*8^2+8+4-1;
G_7(x) = B_8(X_8 +2*8^2+8+3)-1 = X_9 + 2*9^2+9+3-1;
G_8(x) = B_9(X_9 +2*9^2+9+2)-1 = X_10 + 2*10^2+10+2-1 = X_10 + 211;
So finally G_8(x) mod 100 = 211.
The other cases can be proved using the same reasoning.
a(17) = 3.3330...*10^3333, a(18) = 5.555550...*10^555555. - Pontus von Brömssen, Sep 25 2020

			Calculate G_8(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;
G_4(5) = B_5(3*5^3 + 3*5^2 + 3*5 + 2)-1 = 3*6^3 + 3*6^2 + 3*6 + 2-1 = 775;
G_5(5) = B_6(3*6^3 + 3*6^2 + 3*6 + 1)-1 = 3*7^3 + 3*7^2 + 3*7 + 1-1 = 1197;
G_6(5) = B_7(3*7^3 + 3*7^2 + 3*7)-1 = 3*8^3 + 3*8^2 + 3*8-1 = 1751;
G_7(5) = B_8(3*8^3 + 3*8^2 + 2*8 + 7)-1 = 3*9^3 + 3*9^2 + 2*9 + 7-1 = 2454;
G_8(5) = B_9(3*9^3 + 3*9^2 + 2*9 + 6)-1 = 3*10^3 + 3*10^2 + 2*10 + 6-1 = 3325.

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); A271977: G_6(n); A271978: G_7(n); this sequence: 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 A271979(n):
  if n==3: return 0
  for i in range(2,10):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

A271985 G_9(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 253, 4382, 885775, 150051213, 570623341475, 855935016215, 1426559238830, 1997331745490, 3138428376974, 3138428381103, 3138429262496, 3138578427934

Offset: 3

Natan Arie Consigli, Apr 30 2016

nonn

a(17) = 2.066...*10^4574. - Pontus von Brömssen, Sep 25 2020

			Compute G_9(10):
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;
G_9(10) = B_10(5*10^10+5*10^5+5*10^4+5*10^3+5*10^2+5*10+1)-1 = 5*11^11+5*11^5+5*11^4+5*11^3+5*11^2+5*11+1-1 = 1426559238830.

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); A271977: G_6(n); A271978: G_7(n); A271979: G_8(n); this sequence: 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 A271985(n):
  if n==3: return 0
  for i in range(2,11):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

A271986 G_10(n), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

0, 299, 5643, 1357259, 273624711, 17832200896811, 26748301350411, 44580503598539, 62412976762503, 106993205379371, 106993205384715, 106993206736331, 106993479003783

Offset: 3

Natan Arie Consigli, May 01 2016

nonn

a(17) = 1.926...*10^6103. - Pontus von Brömssen, Sep 25 2020

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); A271977: G_6(n); A271978: G_7(n); A271979: G_8(n); A271985: G_9(n); this sequence: 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 A271986(n):
  if n==3: return 0
  for i in range(2,12):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Incorrect program and terms removed by Pontus von Brömssen, Sep 25 2020

A271561 a(n) = G_n(14), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

14, 110, 1281, 18750, 326591, 5862840, 134404971, 3487116548, 100000555551, 3138429262496, 106993206736331, 3937376387710451, 155568095560708189, 6568408355716958693, 295147905179358418247, 14063084452067732533983, 708235345355337686361209, 37589973457545958206423881

Offset: 0

Natan Arie Consigli, Apr 13 2016

			G_1(14) = B_2(14)-1 = B_2(2^(2+1)+2^2+2)-1 = 3^(3+1)+3^3+3-1 = 110;
G_2(14) = B_3(3^(3+1)+3^3+2)-1 = 4^(4+1)+4^4+2-1 = 1281;
G_3(14) = B_4(4^(4+1)+4^4+1)-1 = 5^(5+1)+5^5+1-1 = 18750;
G_4(14) = B_5(5^(5+1)+5^5)-1 = 6^(6+1)+6^6-1 = 326591.

Nicholas Matteo, Table of n, a(n) for n = 0..383

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), A271558: G_n(11), A271559: G_n(12), A271560: G_n(13), A266201: G_n(n).

lista(nn) = {print1(a = 14, ", "); 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, ", "); ); }

A271562 a(n) = G_n(17), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

17, 7625597484987, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084095

Offset: 0

Natan Arie Consigli, Apr 13 2016

			G_1(17) = B_2(17)-1 = B_2(2^2^2+1)-1 = 3^3^3+1-1 = 7625597484987;
G_2(17) = B_3(3^3^3)-1 = 4^4^4-1 has 155 digits;
G_3(17) has 328 digits.

Nicholas Matteo, Table of n, a(n) for n = 0..4

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), A271558: G_n(11), A271559: G_n(12), A271560: G_n(13), A271561: G_n(14), A266201: G_n(n).

A271975 a(n) = G_n(18), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

18, 7625597484989, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Offset: 0

Natan Arie Consigli, Apr 24 2016

			G_1(18) = B_2(18)-1 = B_2(2^2^2+2)-1 = 3^3^3+3-1 = 7625597484989;
G_2(18) = B_3(3^3^3+2)-1 = 4^4^4+2-1 has 154 digits;
G_3(18) = B_4(4^4^4+1)-1 = 5^5^5 has 2184 digits;
G_4(18) = B_5(5^5^5)-1 = 6^6^6-1 = has 36305 digits.

Cf. A215409: G_n(3), A056193: G_n(4), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A271558: G_n(11), A271559: G_n(12), A271560: G_n(13), A271561: G_n(14), A222117: G_n(15), A059933: G_n(16), A271562: G_n(17), A211378: G_n(19), A266201: G_n(n).

A271976 a(n) = G_n(20), where G is the Goodstein function defined in A266201.

Original entry on oeis.org

20, 7625597485013, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084137

Offset: 0

Natan Arie Consigli, Apr 24 2016

			G_1(20) = B_2(20)-1 = B_2(2^2^2+2^2)-1 = 3^3^3+3^3-1 = 7625597485013;
G_2(20) = B_3(3^3^3+2*3^2+2*3+2)-1 = 4^4^4+2*4^2+2*4+2-1  has 154 digits;
G_3(20) = B_4(4^4^4+2*4^2+2*4+1)-1 = 5^5^5+2*5^2+2*5+1-1 has 2184 digits.

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), A271558: G_n(11), A271559: G_n(12), A271560: G_n(13), A271561: G_n(14), A271562: G_n(17), A271975: G_n(18), A266201: G_n(n).

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

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

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A271978 G_7(n), where G is the Goodstein function defined in A266201.

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A271979 G_8(n), where G is the Goodstein function defined in A266201.

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A271985 G_9(n), where G is the Goodstein function defined in A266201.

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A271986 G_10(n), where G is the Goodstein function defined in A266201.

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A271561 a(n) = G_n(14), where G is the Goodstein function defined in A266201.

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A271562 a(n) = G_n(17), where G is the Goodstein function defined in A266201.

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A271975 a(n) = G_n(18), where G is the Goodstein function defined in A266201.

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A271976 a(n) = G_n(20), where G is the Goodstein function defined in A266201.

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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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