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A266202 Weak Goodstein numbers: a(n)= g_n(n), where g_n(n) is the weak Goodstein function. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A nonnegative n in ordinary (depth-1) base-k representation, is n rewritten as 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. Ex. 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: Goodstein's theorem shows that the sequence g_k(n) eventually stabilises and then decreases by 1 in 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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Googology Wiki, Weak Goodstein sequence, see below.

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(PARI) 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

(PARI) 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

CROSSREFS

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;

Sequence in context: A071762 A288945 A245500 * A113721 A127199 A085652

Adjacent sequences:  A266199 A266200 A266201 * A266203 A266204 A266205

KEYWORD

nonn

AUTHOR

Natan Arie' Consigli, Jan 22 2016

EXTENSIONS

More terms from Michel Marcus, Feb 23 2016

STATUS

approved

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Last modified October 13 21:59 EDT 2019. Contains 327981 sequences. (Running on oeis4.)