A059934 Third step in Goodstein sequences, i.e., g(5) if g(2)=n: write g(4)=A057650(n) in hereditary representation base 4, bump to base 5, then subtract 1 to produce g(5).

0, 2, 60, 467, 3125, 3127, 6310, 9842, 15625, 15627, 15685, 16092, 18750, 18752, 53793641718868912174424175024032593379100060

Offset: 2

Author

Henry Bottomley, Feb 12 2001

Keywords

• nonn

Comments

1.911...*10^2184 = a(18) < a(19) < ... < a(31) = a(18) + 18752. - Pontus von Brömssen, Sep 20 2020

Examples

			a(12) = 15685 since with g(2) = 12 = 2^(2 + 1) + 2^2, we get g(3) = 3^(3 + 1) + 3^3-1 = 107 = 3^(3 + 1) + 2*3^2 + 2*3 + 2, g(4) = 4^(4 + 1) + 2*4^2 + 2*4 + 2-1 = 1065 and g(5) = 5^(5 + 1) + 2*5^2 + 2*5^1 + 1-1.
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 2..17
• R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.
• Eric Weisstein's World of Mathematics, Goodstein Sequence
• Wikipedia, Goodstein's Theorem
• Reinhard Zumkeller, Haskell programs for Goodstein sequences

Crossrefs

Cf. A056004, A057650, A059933, A059935, A059936.

Programs

Haskell

-- See Link

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 A059934(n):
  for i in range(2,5):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 20 2020