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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). 16
0, 2, 60, 467, 3125, 3127, 6310, 9842, 15625, 15627, 15685, 16092, 18750, 18752, 53793641718868912174424175024032593379100060 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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

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

EXAMPLE

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.

PROG

(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

CROSSREFS

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

Sequence in context: A048541 A067739 A187626 * A006333 A001760 A230572

Adjacent sequences:  A059931 A059932 A059933 * A059935 A059936 A059937

KEYWORD

nonn

AUTHOR

Henry Bottomley, Feb 12 2001

STATUS

approved

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Last modified January 22 00:57 EST 2021. Contains 340359 sequences. (Running on oeis4.)