A267648 -id:A267648 - 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

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.

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}]

A271991 g_n(10) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

10, 29, 65, 125, 215, 284, 363, 452, 551, 660, 779, 907, 1045, 1193, 1351, 1519, 1697, 1885, 2083, 2291, 2509, 2737, 2975, 3222, 3479, 3746, 4023, 4310, 4607, 4914, 5231, 5558, 5895, 6242, 6599, 6966, 7343, 7730, 8127, 8534, 8951

Offset: 0

Natan Arie Consigli, May 22 2016

For more info see A266201-A266202.

Cf. A271557: G_n(10).

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); A271989: g_n(8); A271990: g_n(9); 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, 10], {n, 0, 40}]

A271992 g_n(16) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

16, 80, 169, 310, 515, 795, 1163, 1631, 2211, 2915, 3755, 4742, 5889, 7208, 8711, 10410, 12317, 14444, 16803, 19406, 22265, 25392, 28799, 32472, 36447, 40736, 45351, 50304, 55607, 61272, 67311, 73736, 80559, 87792, 95447, 103536, 112071

Offset: 0

Natan Arie Consigli, May 24 2016

For more information see A266201 and A266202.

Cf. A271557: G_n(10).

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); A265034: g_n(266); 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, 16], {n, 0, 36}]

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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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.

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

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

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

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

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