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

Author

Natan Arie Consigli, Jan 22 2016

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

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.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Feb 23 2016