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A059935 Fourth step in Goodstein sequences, i.e., g(6) if g(2)=n: write g(5)=A059934(n) in hereditary representation base 5, bump to base 6, then subtract 1 to produce g(6).

%I A059935 #21 Feb 16 2025 08:32:44
%S A059935 1,83,775,46655,46657,93395,140743,279935,279937,280019,280711,326591,
%T A059935 326593,
%U A059935 19916489515870532960258562190639398471599239042185934648024761145811
%N A059935 Fourth step in Goodstein sequences, i.e., g(6) if g(2)=n: write g(5)=A059934(n) in hereditary representation base 5, bump to base 6, then subtract 1 to produce g(6).
%C A059935 2.659...*10^36305 = a(18) < a(19) < ... < a(31) = a(18) + 326594. - _Pontus von Brömssen_, Sep 20 2020
%H A059935 Pontus von Brömssen, <a href="/A059935/b059935.txt">Table of n, a(n) for n = 3..17</a>
%H A059935 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 A059935 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence</a>
%H A059935 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>
%H A059935 Reinhard Zumkeller, <a href="/A211378/a211378.hs.txt">Haskell programs for Goodstein sequences</a>
%e A059935 a(12) = 280019 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 and g(6) = 6^(6 + 1) + 2*6^2 + 6 + 5 = 280019.
%o A059935 (Haskell) -- See Link
%o A059935 (Python)
%o A059935 from sympy.ntheory.factor_ import digits
%o A059935 def bump(n,b):
%o A059935   s=digits(n,b)[1:]
%o A059935   l=len(s)
%o A059935   return sum(s[i]*(b+1)**bump(l-i-1,b) for i in range(l) if s[i])
%o A059935 def A059935(n):
%o A059935   for i in range(2,6):
%o A059935     n=bump(n,i)-1
%o A059935   return n # _Pontus von Brömssen_, Sep 20 2020
%Y A059935 Cf. A056004, A057650, A059933, A059934, A059936.
%K A059935 nonn
%O A059935 3,2
%A A059935 _Henry Bottomley_, Feb 12 2001