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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).

%I A059936 #28 Feb 16 2025 08:32:44
%S A059936 0,109,1197,98039,823543,1647195,2471826,4215754,5764801,5764910,
%T A059936 5765998,5862840,6588344,
%U A059936 5103708485122940631839901111036829791435007685667303872450435153015345686896530517814322070729709
%N 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).
%C A059936 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
%H A059936 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 A059936 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence</a>
%H A059936 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>
%H A059936 Reinhard Zumkeller, <a href="/A211378/a211378.hs.txt">Haskell programs for Goodstein sequences</a>
%e A059936 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.
%o A059936 (Haskell) -- See Link
%o A059936 (Python)
%o A059936 from sympy.ntheory.factor_ import digits
%o A059936 def bump(n,b):
%o A059936   s=digits(n,b)[1:]
%o A059936   l=len(s)
%o A059936   return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
%o A059936 def A059936(n):
%o A059936   for i in range(2,7):
%o A059936     n=bump(n,i)-1
%o A059936   return n # _Pontus von Brömssen_, Sep 19 2020
%Y A059936 Cf. A056004, A057650, A059933, A059934, A059935.
%K A059936 nonn
%O A059936 3,2
%A A059936 _Henry Bottomley_, Feb 12 2001
%E A059936 a(16) corrected by _Pontus von Brömssen_, Sep 18 2020