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Digits of the 8-adic integer (1/7)^(1/7).
3

%I #17 Aug 17 2019 05:06:45

%S 7,0,5,4,4,7,7,4,1,6,1,3,3,0,3,4,5,0,5,4,2,7,5,3,1,4,7,6,0,6,1,2,4,6,

%T 2,2,1,6,2,0,2,5,1,6,3,4,0,6,1,2,4,0,5,6,5,5,0,4,6,7,5,4,0,0,1,6,3,6,

%U 7,6,1,2,7,2,3,3,7,1,5,5,4,6,3,4,6,1,3,3,3,2,6,1,4,3,0,0,1,4,4,5,4

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

%H Seiichi Manyama, <a href="/A309724/b309724.txt">Table of n, a(n) for n = 0..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hensel%27s_lemma">Hensel's Lemma</a>.

%F Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + 7 * (7 * b(n-1)^7 - 1) mod 8^n for n > 1, then a(n) = (b(n+1) - b(n))/8^n.

%o (PARI) N=100; Vecrev(digits(lift((1/7+O(2^(3*N)))^(1/7)), 8), N)

%o (Ruby)

%o def A309724(n)

%o ary = [7]

%o a = 7

%o n.times{|i|

%o b = (a + 7 * (7 * a ** 7 - 1)) % (8 ** (i + 2))

%o ary << (b - a) / (8 ** (i + 1))

%o a = b

%o }

%o ary

%o end

%o p A309724(100)

%Y Digits of the k-adic integer (1/(k-1))^(1/(k-1)): A309722 (k=4), A309723 (k=6), this sequence (k=8), A225464 (k=10).

%Y Cf. A309700.

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Aug 14 2019