A271977 G_6(n), where G is the Goodstein function defined in A266201.

0, 139, 1751, 187243, 16777215, 33554571, 50333399, 84073323, 134217727, 134217867, 134219479, 134404971, 150994943

Offset: 3

Author

Natan Arie Consigli, Apr 24 2016

Keywords

• nonn

Comments

The next term (line break for better formatting) is a(16) = \

1619239197880733074062994004113160848331305687934176134326809 \

538279709713884753268291640071900343455846003089194770060104834018705547.

a(17) = 2.870...*10^1585, a(18) = 6.943...*10^169099. - Pontus von Brömssen, Sep 24 2020

Examples

			Find G_6(7):
G_1(7) = B_2(7)-1= B_2(2^2+2+1)-1 = 3^3+3+1-1 = 30;
G_2(7) = B_3(G_1(7))-1 = B_3(3^3+3)-1 =  4^4+4-1 = 259;
G_3(7) = B_4(G_2(7))-1 = 5^5+3-1 = 3127;
G_4(7) = B_5(G_3(7))-1 = 6^6+2-1 = 46657;
G_5(7) = B_6(G_4(7))-1 = 7^7+1-1 = 823543;
G_6(7) = B_7(G_5(7))-1 = 8^8-1 = 16777215.
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 3..16
• Wikipedia, Goodstein's theorem

Crossrefs

Cf. A056004: G_1(n); A057650: G_2(n); A059934: G_3(n); A059935: G_4(n); A059936: G_5(n); this sequence: G_6(n); A271978: G_7(n); A271979: G_8(n); A271985: G_9(n); A271986: G_10(n); A266201: G_n(n).

Programs

Python

from sympy.ntheory.factor_ import digits
def bump(n,b):
  s=digits(n,b)[1:]
  l=len(s)
  return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
def A271977(n):
  if n==3: return 0
  for i in range(2,8):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 24 2020

Extensions

a(10) corrected by Pontus von Brömssen, Sep 24 2020