A271978 G_7(n), where G is the Goodstein function defined in A266201.

0, 173, 2454, 332147, 37665879, 774841151, 1162263921, 1937434592, 2749609302, 3486784574, 3486786855, 3487116548, 3524450280

Offset: 3

Author

Natan Arie Consigli, Apr 30 2016

Keywords

• nonn

Comments

a(16) is too big to include - see b-file. a(17) = 9.221...*10^2347, a(18) = 2.509...*10^316952. - Pontus von Brömssen, Sep 25 2020

Examples

			Find G_7(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;
G_7(7) = B_8(G_6(7))-1 = 7*9^7+7*9^6+7*9^5+7*9^4+7*9^3+7*9^2+7*9+7-1 = 37665879.
		

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); A271977: G_6(n); this sequence: 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 A271978(n):
  if n==3: return 0
  for i in range(2,9):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 25 2020

Extensions

a(9) corrected by Pontus von Brömssen, Sep 25 2020