A266201 -id:A266201 - cp's OEIS frontend

A266202 Weak Goodstein numbers: a(n) = g_n(n), where g_n(n) is the weak Goodstein function.

0, 0, 1, 2, 11, 21, 43, 69, 211, 389, 779, 1276, 2753, 3405, 4167, 5029, 12317, 21691, 42083, 68050, 234257, 279872, 331871, 390781, 458271, 533659, 618679, 713344, 831407, 953343, 1081455, 1222053, 2753231, 4634203, 8637959, 13483492, 49254279, 90224223, 102400127

Offset: 0

Natan Arie Consigli, Jan 22 2016

nonn

A nonnegative n in ordinary (depth-1) base-k representation is n rewritten as a linear combination k powers n = n_1*b^m_1 + ... + n_k*b^m_k where 0 < n_i < b and m_1 > ... > m_k >= 0.
For instance, the ordinary representation of 34 in base 3 is 3^3 + 2*3 + 1.
Let b_k(n) be the function that substitutes the bases of the base-k representation of n with the base k+1. E.g., b_3(34) = b_3(3^3 + 2*3 + 1) = 4^3 + 2*4 + 1 = 73.
Define the weak Goodstein function as: g_k(n) = b_(k+1)(g_(k-1)(n))-1, g_0(n) = n.
See example for instances.
Let n be a fixed nonnegative integer: Goodstein's theorem shows that the sequence g_k(n) eventually stabilizes and then decreases by 1 at each step until it reaches 0. Thereafter, all the values of g_k(n) < 0 are not part of the sequence.
By Goodstein's theorem we conclude that g_k(n) is a finite sequence.

			Find a(5) = g_5(5):
g_0(5) = 5;
g_1(5) = b_2(5)-1 = b_2(2^2+1)-1 = 3^2+1-1 = 9;
g_2(5) = b_3(3^2)-1 = 4^2-1 = 15;
g_3(5) = b_4(3*4 + 3)-1 = 3*5+3-1 = 17;
g_4(5) = b_5(3*5 + 2)-1 = 3*6 + 2-1 = 19;
g_5(5) = b_6(3*6 + 1)-1 = 3*7+1-1 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Googology Wiki, Weak Goodstein sequence, see below.

Cf. A266201 ("Strong" Goodstein numbers).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A266203: a(n) = k such that g_k(n)=0.

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, n], {n, 0, 38}] (* Michael De Vlieger, Mar 18 2016 *)

a(n) = {if (n == 0, return (0)); wn = n; for (k=2, n+1, pd = Pol(digits(wn, k)); wn = subst(pd, x, k+1) - 1;); wn;} \\ Michel Marcus, Feb 23 2016

a(n) = {if (n == 0, return (0)); wn = n; for(k=2, n+1, vd = digits(wn, k); wn = fromdigits(vd, k+1) - 1;); wn;} \\ Michel Marcus, Feb 19 2017

More terms from Michel Marcus, Feb 23 2016

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

A267647 a(n) = g_n(4), where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

4, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Natan Arie Consigli, Mar 17 2016

For more info see A266201 - A266202.

			g_1(4) = b_2(4)-1 = b_2(2^2)-1 = 3^2-1 = 8;
g_2(4) = b_3(2*3+2)-1 = 2*4 + 2-1 = 9;
g_3(4) = b_4(2*4+1)-1 = 2*5 + 1-1 = 10;
g_4(4) = b_5(2*5)-1= 2*6 - 1 = 11;
g_5(4) = b_6(6+5)-1 = 7+5-1 = 11;
g_6(4) = b_7(7+4)-1 = 8+4-1 = 11;
g_7(4) = b_8(8+3)-1 = 9+3-1 = 11;
g_8(4) = b_9(9+2)-1 = 10+2-1 = 11;
g_9(4) = b_10(10+1)-1 = 11+1-1 = 11;
g_10(4) = b_11(11)-1 = 12-1 = 11;
g_11(4) = b_12(11)-1 = 11-1 = 10;
g_12(4) = b_13(10)-1 = 10-1 = 9;
g_13(4) = b_14(9)-1 = 9-1 = 8;
…
g_21(4) = 0;

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A266202: g_n(n); A267648: g_5(n); A266203: a(n) = k such that g_k(n)=0;

A056193: G_n(4).

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 4], {n, 0, 21}] (* Michael De Vlieger, Mar 18 2016 *)

a(n) = {if (n == 0, return (4)); wn = 4; for (k=2, n+1, pd = Pol(digits(wn, k)); wn = subst(pd, x, k+1) - 1; ); wn; }
vector(22, n, n--; a(n)) \\ Michel Marcus, Apr 03 2016

A267648 a(n) = g_n(5) where g is the function defined in A266202.

Original entry on oeis.org

5, 9, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Natan Arie Consigli, Mar 17 2016

For more info see A266201-A266202.

This sequence has A266203(5)+1=62 terms and a(A266203(5))=a(61)=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(5)-1)/2)=a(30)=(A266203(5)+1)/2=31. - Zhuorui He, Aug 08 2025

			g_1(5) = b_2(5)-1 = b_2(2^2+1)-1 = 3^2+1-1 = 9;
g_2(5) = b_3(3^2)-1 = 4^2-1 = 15;
g_3(5) = b_4(3*4+3)-1 = 3*5+3-1 = 17;
g_4(5) = b_5(3*5 + 2)-1 = 3*6 + 2-1 = 19;
g_5(5) = b_6(3*6 + 1)-1 = 3*7+1-1 = 21;
g_6(5) = b_7(3*7)-1 = 3*8-1 = 23;
g_7(5) = b_8(2*8+7)-1 = 2*9+7-1 = 24;
g_8(5) = b_9(2*9+6)-1 = 2*10+6-1 = 25;
g_9(5) = b_10(2*10+5)-1 = 2*11+5-1 = 26;
g_10(5) = b_11(2*11+4)-1 = 2*12+4-1 = 27;
g_11(5) = b_12(2*12+3)-1 = 2*13+3-1 = 28;
g_12(5) = b_13(2*13+2)-1 = 2*14+2-1 = 29;
g_13(5) = b_14(2*14+1)-1 = 2*15+1-1 = 30;
g_14(5) = b_15(2*15)-1 = 2*16-1 = 31;
g_15(5) = b_16(16+15)-1 = 17+15-1 = 31;
...
g_30(5) = b_31(31)-1 = 31;
g_31(5) = b_32(31)-1 = 30;
g_32(5) = b_33(30)-1 = 29;
...
g_61(5) = 0. (End of sequence)

Cf. A266204: G_n(5).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A266202: g_n(n); A266203: a(n) = k such that g_k(n)=0.

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 5], {n, 0, 61}] (* Michael De Vlieger, May 17 2016 *)

a(n, m=5) = { my(wn = m); for (k=2, n+1, wn = fromdigits(digits(wn, k), k+1) - 1); wn; }
vector(62, n, n--; a(n)) \\ Michel Marcus, Apr 03 2016 and Aug 08 2025

Duplicated a(31) removed by Zhuorui He, Aug 07 2025

A046859 Simplified Ackermann function (main diagonal of Ackermann-Péter function).

Original entry on oeis.org

1, 3, 7, 61

Offset: 0

Don Knuth

The next term is 2^(2^(2^(2^16))) - 3, which is too large to display in the DATA lines.
Another version of the Ackermann numbers is the sequence 1^1, 2^^2, 3^^^3, 4^^^^4, 5^^^^^5, ..., which begins 1, 4, 3^3^3^... (where the number of 3's in the tower is 3^3^3 = 7625597484987), ... [Conway and Guy]. This grows too rapidly to have its own entry in the OEIS.
An even more rapidly growing sequence is the Conway-Guy sequence 1, 2->2, 3->3->3, 4->4->4->4, ..., which agrees with the sequence in the previous comment for n <= 3, but then the 4th term is very much larger than 4^^^^4.
From Natan Arie Consigli, Apr 10 2016: (Start)
A189896 = succ(0), 1+1, 2*2, 3^3,..., also called Ackermann numbers, is a weaker version of the above sequence.
The Ackermann functions are well-known to be simple examples of computable (implementable using a combination of while/for-loops) but not primitive recursive (implementable using only for-loops) functions.
See A054871 for the definitions of the hyperoperations (a[n]b and H_n(a,b)).
The original Ackermann function f is defined by:
{
{f(0,y,z)=y+z;
{f(1,y,0)=0;
{f(2,y,0)=1;
{f(x,y,0)=x;
{f(x,y,z)=f(x-1,y,f(x,y,z-1))
{
Here we have f(1,y,z)=y*z, f(2,y,z)=y^z.
Ackermann function variants are 3-argument functions that satisfy the recurrence relation above.
Example:
the hyperoperation function H(x,y,z) satisfies the original's recurrence relation but has the following initial values:
{
{H(0,y,z) = y+1;
{H(1,y,0) = y;
{H(2,y,0) = 0;
{H(n,y,0) = 1.
{
The family of Ackermann functions can be simplified by omitting the "y" variable of the 3-argument function by making them have two arguments.
A 2-argument Ackermann function would then be a function satisfying the recurrence relation: f(x,z)=f(x-1,f(x,z-1)).
The most popular example is Ackermann-Péter's function defined by:
{
{A(0,y) = y+1;
{A(x+1,0) = A(x,1);
{A(x+1,y+1) = A(x,A(x+1,y))
{
Here we have A(0,y-1) = y = 2[0](y-1+3)-3.
Suppose A(x-1,y-1) = 2[x-1](y-1+3)-3.
By induction on positive x:
since 2[x]2 = 4 (See A255176) we have A(x,0) = A(x-1,1) = 2[x-1]4-3 = 2[x-1]2[x-1]2-3 = 2[x-1]3-3.
By induction on positive y we can conclude that:
A(x,y) = A(x-1,A(x,y-1)) = 2[x-1](2[x](y-1+3)-3+3)-3 = 2[x-1]2[x](y-1+3)-3 = 2[x](y+3)-3.
*
If f is a 3-argument (2-argument) Ackermann function, Ack(n) = f(n,n,n) (f(n,n)) is called a simplified Ackermann function. The "Ackermann numbers" are the values of Ack(n).
Here we have a(n) = A(n,n) = 2[n](n+3)-3.
(End)

			From _Natan Arie Consigli_, Apr 10 2016: (Start)
a(0) = 2[0](0+3)-3 = 1;
a(1) = 2[1](1+3)-3 = 3;
a(2) = 2[2](2+3)-3 = 7;
a(3) = 2[3](3+3)-3 = 61;
a(4) = 2[4](4+3)-3 = 2^(2^(2^65536)) - 3.  (End)

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 60, 1996.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
H. Hermes, Aufzaehlbarkeit, Entscheidbarkeit, Berechenbarkeit: Einfuehrung in die Theorie der rekursiven Funktionen (3rd ed., Springer, 1978), 83-89.
H. Hermes, ditto, 2nd ed. also available in English (Springer, 1969), ch. 13

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
D. E. Knuth and N. J. A. Sloane, Correspondence, May 1970
Index entries for sequences related to Ackermann function

Cf. A059936, A266200, A271553. (sequences involving simplified Ackermann Functions)
Cf. A001695, A014221, A143797, A264929 (sequences involving other versions of two-argument Ackermann's Function).
Cf. A054871, A189896 (sequences involving variants of the three-argument Ackermann's Function).
Cf. A126333 (a(n)=A(n,0)), A074877 (a(n)=A(3,n)).
Cf. A260002-A260006 (sequences with Sudan's function, another computable but not primitive recursive function).
Cf. A266201 (Goodstein's function, total and not primitive recursive).

From Natan Arie Consigli, Apr 10 2016: (Start)
A(0, y) := y+1, A(x+1, 0) := A(x, 1), A(x+1, y+1) := A(x, A(x+1, y));
a(n) = A(n,n).
a(n) = 2[n](n+3)-3 = H_n(2,n+3)-3.  (End)

Additional comments from Frank Ellermann, Apr 21 2001

Name clarified by Natan Arie Consigli, May 13 2016

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

A271987 g_n(6) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

6, 11, 17, 25, 35, 39, 43, 47, 51, 55, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161

Offset: 0

Natan Arie Consigli, May 15 2016

For more info see A266201-A266202.

			g_1(6) = b_2(6)-1 = b_2(2^2+2)-1 = 3^2+3-1 = 11;
g_2(6) = b_3(3^2+2)-1 = 4^2+2-1 = 17;
g_3(6) = b_4(4^2+1)-1 = 5^2+1-1 = 25;
g_4(6) = b_5(5^2)-1 = 6^2-1 = 35;
g_5(6) = b_6(5*6+5)-1 = 5*7+5-1 = 39;
g_6(6) = b_7(5*7+4)-1 = 5*8+4-1 = 43;
g_7(6) = b_8(5*8+3)-1 = 5*9+3-1 = 47;
g_8(6) = b_9(5*9+2)-1 = 5*10+2-1 = 51;
g_9(6) = b_10( 5*10+1)-1 = 5*11+1-1= 55;
g_10(6) = b_11(5*11)-1 = 5*12-1 = 59;
g_11(6) = b_12(4*12+11)-1 = 4*13+11-1= 62;
g_12(6) = b_13(4*13+10)-1 = 4*14+10-1 = 65;
...
g_381(6) = 0.

Michael De Vlieger, Table of n, a(n) for n = 0..381

Cf. A266205: G_n(6).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 6], {n, 0, 64}] (* Michael De Vlieger, May 17 2016 *)

A271988 g_n(7) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

7, 12, 19, 27, 37, 49, 63, 69, 75, 81, 87, 93, 99, 105, 111, 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, 176, 181, 186, 191, 195, 199, 203, 207, 211, 215, 219, 223, 227, 231, 235, 239, 243, 247, 251, 255, 259, 263, 267, 271, 275, 279, 283, 287, 291, 295, 299, 303, 307, 311, 315, 319, 322, 325

Offset: 0

Natan Arie Consigli, May 21 2016

For more info see A266201-A266202.

			g_1(7)= b_2(7)-1 = b_2(2^2+2+1)-1 = 3^2+3+1-1 = 12;
g_2(7) = b_3(3^2+3)-1 = 4^2+4-1 = 19;
g_3(7) = b_4(4^2+3)-1 = 5^2+3-1 = 27;
g_4(7) = b_5(5^2+2)-1 = 6^2+2-1 = 37;
g_5(7) = b_6(6^2+1)-1 = 7^2+1-1 = 49;
g_6(7) = b_7(7^2)-1 = 8^2-1 = 63;
g_7(7) = b_8(7*8+7)-1 = 7*9+7-1 = 69;
...
g_2045(7) = 0.

Cf. A271554: G_n(7).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 7], {n, 0, 64}]

A271989 g_n(8) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

8, 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, 3525, 3645, 3767, 3891, 4017, 4145, 4275, 4407, 4541

Offset: 0

Natan Arie Consigli, May 22 2016

For more info see A266201-A266202.

			g_1(8) = b_2(8)-1 = b_2(2^3)-1 = 3^3-1 = 26;
g_2(8) = b_3(2*3^2+2*3+2)-1 = 2*4^2+2*4+2-1 = 41;
g_3(8) = b_4(2*4^2+2*4+1)-1 = 2*5^2+2*5+1-1 = 60;
g_4(8) = b_5(2*5^2+2*5)-1 = 2*6^2+2*6-1 = 83;
g_5(8) = b_6(2*6^2+6+5)-1 = 2*7^2+7+5-1 = 109;
g_6(8) = b_7(2*7^2+7+4)-1 = 2*8^2+8+4-1 = 139;
g_7(8) = b_8(2*8^2+8+3)-1 = 2*9^2+9+3-1 = 173;
g_8(8) = b_9(2*9^2+9+2)-1 = 2*10^2+10+2-1 = 211;
g_9(8) = b_10(2*10^2+10+1)-1 = 2*11^2+11+1-1 = 253;
g_10(8) = b_11(2*11^2+11)-1 = 2*12^2+12-1 = 299.

Essentially the same as A056193.
Cf. G_n(8): A271555.
Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 8], {n, 0, 55}]

A271990 g_n(9) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

9, 27, 63, 92, 127, 168, 215, 267, 325, 389, 459, 535, 617, 705, 799, 898, 1003, 1114, 1231, 1354, 1483, 1618, 1759, 1906, 2059, 2218, 2383, 2554, 2731, 2914, 3103, 3297, 3497, 3703, 3915, 4133, 4357, 4587, 4823, 5065, 5313, 5567, 5827, 6093, 6365, 6643

Offset: 0

Natan Arie Consigli, May 22 2016

For more info see A266201-A266202.

Cf. A271556: G_n(9).

Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 9], {n, 0, 45}]

cp's OEIS Frontend

A266202 Weak Goodstein numbers: a(n) = g_n(n), where g_n(n) is the weak Goodstein function.

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A137411 Weak Goodstein sequence starting at 11.

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A267647 a(n) = g_n(4), where g is the weak Goodstein function defined in A266202.

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A267648 a(n) = g_n(5) where g is the function defined in A266202.

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A046859 Simplified Ackermann function (main diagonal of Ackermann-Péter function).

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

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A271987 g_n(6) where g is the weak Goodstein function defined in A266202.

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