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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

%I

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

%T 53793641718868912174424175024032593379100060

%N 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).

%C 1.911...*10^2184 = a(18) < a(19) < ... < a(31) = a(18) + 18752. - _Pontus von Brömssen_, Sep 20 2020

%H Pontus von Brömssen, <a href="/A059934/b059934.txt">Table of n, a(n) for n = 2..17</a>

%H R. L. Goodstein, <a href="http://www.jstor.org/stable/2268019">On the Restricted Ordinal Theorem</a>, J. Symb. Logic 9, 33-41, 1944.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>

%H Reinhard Zumkeller, <a href="/A211378/a211378.hs.txt">Haskell programs for Goodstein sequences</a>

%e 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.

%o (Haskell) see Link

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o def bump(n,b):

%o s=digits(n,b)[1:]

%o l=len(s)

%o return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])

%o def A059934(n):

%o for i in range(2,5):

%o n=bump(n,i)-1

%o return n # _Pontus von Brömssen_, Sep 20 2020

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

%K nonn

%O 2,2

%A _Henry Bottomley_, Feb 12 2001

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Last modified March 2 19:18 EST 2021. Contains 341756 sequences. (Running on oeis4.)