A059936 Fifth step in Goodstein sequences, i.e., g(7) if g(2)=n: write g(6)=A059935(n) in hereditary representation base 6, bump to base 7, then subtract 1 to produce g(7).

0, 109, 1197, 98039, 823543, 1647195, 2471826, 4215754, 5764801, 5764910, 5765998, 5862840, 6588344, 5103708485122940631839901111036829791435007685667303872450435153015345686896530517814322070729709

Offset: 3

Author

Henry Bottomley, Feb 12 2001

Keywords

• nonn

Comments

a(17) = 4.587...*10^1014, a(18) = 1.505...*10^82854, and 3.759...*10^695974 = a(19) < a(20) < ... < a(31) = a(19) + 6588344. - Pontus von Brömssen, Sep 20 2020

Examples

			a(12) = 5764910 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 + 1 = 1065, g(5) = 5^(5 + 1) + 2*5^2 + 2*5 = 15685, g(6) = 6^(6 + 1) + 2*6^2 + 6 + 5 = 280019 and g(7) = 7^(7 + 1) + 2*7^2 + 7 + 4 = 5764910.
		

Links

• 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, A059934, A059935.

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 A059936(n):
  for i in range(2,7):
    n=bump(n,i)-1
  return n # Pontus von Brömssen, Sep 19 2020

Extensions

a(16) corrected by Pontus von Brömssen, Sep 18 2020