A059933 -id:A059933 - cp's OEIS frontend

A059934 Third step in Goodstein sequences, i.e., g(5) if g(2)=n: write g(4)=A057650(n) in hereditary representation base 4, bump to base 5, then subtract 1 to produce g(5).

0, 2, 60, 467, 3125, 3127, 6310, 9842, 15625, 15627, 15685, 16092, 18750, 18752, 53793641718868912174424175024032593379100060

Offset: 2

Henry Bottomley, Feb 12 2001

nonn

1.911...*10^2184 = a(18) < a(19) < ... < a(31) = a(18) + 18752. - Pontus von Brömssen, Sep 20 2020

			a(12) = 15685 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 + 2-1 = 1065 and g(5) = 5^(5 + 1) + 2*5^2 + 2*5^1 + 1-1.

Pontus von Brömssen, Table of n, a(n) for n = 2..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, A059935, 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 A059934(n):
  for i in range(2,5):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 20 2020

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).

Original entry on oeis.org

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

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, ", "); ); }

A137411 Weak Goodstein sequence starting at 11.

Original entry on oeis.org

11, 30, 67, 127, 217, 343, 511, 636, 775, 928, 1095, 1276, 1471, 1680, 1903, 2139, 2389, 2653, 2931, 3223, 3529, 3849, 4183, 4531, 4893, 5269, 5659, 6063, 6481, 6913, 7359, 7818, 8291, 8778, 9279, 9794, 10323, 10866, 11423, 11994, 12579, 13178

Offset: 0

Nicholas Matteo (kundor(AT)kundor.org), Apr 15 2008

The sequence eventually goes to zero, as can be seen by noting that multiples of the highest exponent (3 in this case) only go down; in fact the 8th term, a(8) = 7*8^2 + 7*8 + 7 = 511; after which the multiple of the square term will only go down, etc.
This sequence, for 11, grows beyond the quintillions of digits before going to zero.
From Zhuorui He, Aug 07 2025: (Start)
For more info see A266201-A266202.
This sequence has A266203(11)+1 terms and a(A266203(11))=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(11)-1)/2)=(A266203(11)+1)/2. 10^^8 < A266203(11) < 10^^9.
More precisely, 10^(10^(10^(10^(10^(10^(10^619.29937)))))) < A266203(11) < 10^(10^(10^(10^(10^(10^(10^619.299371)))))). (End)

			a(0) = 11 = 2^3 + 2^1 + 2^0
a(1) = 3^3 + 3^1 + 3^0 - 1 = 30
a(2) = 4^3 + 4^1 - 1 = 4^3 + 3*4^0 = 67

K. Hrbacek and T. Jech, Introduction to Set Theory, Taylor & Francis Group, 1999, pp. 125-127.

Zhuorui He, Table of n, a(n) for n = 0..10000 (Terms 0..998 from Harvey P. Dale)

Cf. A056004 (strong Goodstein sequences), A059933 (strong Goodstein sequence for 16.).

Weak Goodstein sequences: A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A271989: g_n(8); A271990: g_n(9); A271991: g_n(10); A137411: g_n(11); A271992: g_n(16); A265034: g_n(266); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

nxt[{n_,a_}]:={n+1,FromDigits[IntegerDigits[a,n+1],n+2]-1}; Transpose[ NestList[ nxt,{1,11},50]][[2]] (* Harvey P. Dale, Feb 09 2015 *)

a(n, m=11) = { my(wn = m); for (k=2, n+1, wn = fromdigits(digits(wn, k), k+1) - 1); wn; } \\ Zhuorui He, Aug 08 2025

To obtain a(n + 1), write a(n) in base n + 2, increase the base to n + 3 and subtract 1.

Offset changed to 0 by Zhuorui He, Aug 07 2025

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

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, ", "); ); }

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.

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

cp's OEIS Frontend

A059934 Third step in Goodstein sequences, i.e., g(5) if g(2)=n: write g(4)=A057650(n) in hereditary representation base 4, bump to base 5, then subtract 1 to produce g(5).

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

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A137411 Weak Goodstein sequence starting at 11.

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A265034 Weak Goodstein sequence beginning with 266.

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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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