A266202 -id:A266202 - cp's OEIS frontend

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

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

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}]

A268687 a(n) = MAX(g_k(n)) where g_k(n) is the function defined in A266202.

Original entry on oeis.org

0, 1, 2, 3, 11, 31, 191, 1023

Offset: 0

Natan Arie Consigli, Apr 02 2016

			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;
So a(4)=11.

Cf. A266203, A268688, A268689.

g(n, k) = {if (n == 0, return (k)); wn = k; for (k=2, n+1, pd = Pol(digits(wn, k)); wn = subst(pd, x, k+1) - 1; ); wn; }
a(n) = {vg = []; ok = 1; ns = 0; while(ok, newg = g(ns, n); vg = concat(vg, newg); if (newg <= 0, ok = 0); ns++;); vmax = vecmax(vg); vmax;} \\ Michel Marcus, Apr 04 2016; corrected Jun 13 2022

a(6)-a(7) from Michel Marcus, Apr 04 2016

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}]

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

cp's OEIS Frontend

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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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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A268687 a(n) = MAX(g_k(n)) where g_k(n) is the function defined in A266202.

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

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

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