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A309700 Digits of the 8-adic integer 7^(1/7).

%I A309700 #20 Aug 14 2019 08:32:03
%S A309700 7,6,1,0,1,6,4,1,7,3,6,4,4,5,3,3,4,2,0,0,6,2,5,4,2,6,6,3,2,2,6,1,0,3,
%T A309700 5,6,1,6,6,7,0,6,6,7,7,5,3,2,2,7,5,5,1,7,5,7,1,1,1,2,5,0,4,3,2,5,3,0,
%U A309700 3,3,1,7,3,4,5,4,5,1,1,2,2,7,0,6,7,1,4,4,6,7,6,2,2,5
%N A309700 Digits of the 8-adic integer 7^(1/7).
%H A309700 Seiichi Manyama, <a href="/A309700/b309700.txt">Table of n, a(n) for n = 0..10000</a>
%H A309700 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.
%F A309700 Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + b(n-1)^7 - 7 mod 8^n for n > 1, then a(n) = (b(n+1) - b(n))/8^n.
%o A309700 (PARI) N=100; Vecrev(digits(lift((7+O(2^(3*N)))^(1/7)), 8), N)
%o A309700 (Ruby)
%o A309700 def A309700(n)
%o A309700   ary = [7]
%o A309700   a = 7
%o A309700   n.times{|i|
%o A309700     b = (a + a ** 7 - 7) % (8 ** (i + 2))
%o A309700     ary << (b - a) / (8 ** (i + 1))
%o A309700     a = b
%o A309700   }
%o A309700   ary
%o A309700 end
%o A309700 p A309700(100)
%Y A309700 Digits of the k-adic integer (k-1)^(1/(k-1)): A309698 (k=4), A309699 (k=6), this sequence (k=8), A225458 (k=10).
%Y A309700 Cf. A225445.
%K A309700 nonn,base
%O A309700 0,1
%A A309700 _Seiichi Manyama_, Aug 13 2019