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A309699 Digits of the 6-adic integer 5^(1/5).

%I A309699 #22 Aug 14 2019 08:31:52
%S A309699 5,4,0,3,1,5,0,0,3,3,2,1,3,0,0,3,4,3,1,1,1,1,1,4,3,4,0,5,3,1,1,5,3,3,
%T A309699 0,2,2,2,5,3,5,5,2,5,2,2,2,3,4,2,0,5,4,3,3,2,0,0,4,1,1,5,5,5,0,0,1,4,
%U A309699 3,5,4,5,1,5,5,0,5,4,0,4,4,4,4,3,4,4,0,4,3,4,0,5,4,4
%N A309699 Digits of the 6-adic integer 5^(1/5).
%C A309699 x   = ...513045,
%C A309699 x^2 = ...433521,
%C A309699 x^3 = ...051525,
%C A309699 x^4 = ...354241,
%C A309699 x^5 = ...000005.
%H A309699 Seiichi Manyama, <a href="/A309699/b309699.txt">Table of n, a(n) for n = 0..10000</a>
%H A309699 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A309699 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 5, b(n) = b(n-1) + b(n-1)^5 - 5 mod 6^n for n > 1, then a(n) = (b(n+1) - b(n))/6^n.
%o A309699 (PARI) N=100; Vecrev(digits(lift(chinese(Mod((5+O(2^N))^(1/5), 2^N), Mod((5+O(3^N))^(1/5), 3^N))), 6), N)
%o A309699 (Ruby)
%o A309699 def A309699(n)
%o A309699   ary = [5]
%o A309699   a = 5
%o A309699   n.times{|i|
%o A309699     b = (a + a ** 5 - 5) % (6 ** (i + 2))
%o A309699     ary << (b - a) / (6 ** (i + 1))
%o A309699     a = b
%o A309699   }
%o A309699   ary
%o A309699 end
%o A309699 p A309699(100)
%Y A309699 Digits of the k-adic integer (k-1)^(1/(k-1)): A309698 (k=4), this sequence (k=6), A309700 (k=8), A225458 (k=10).
%Y A309699 Cf. A309448.
%K A309699 nonn,base
%O A309699 0,1
%A A309699 _Seiichi Manyama_, Aug 13 2019