%I #16 Aug 07 2017 09:11:53

%S 0,3,3,53,303,2178,5303,67803,67803,849053,6708428,6708428,6708428,

%T 983270928,983270928,25397333428,147467645928,605231317803,

%U 1368170770928,5182868036553,43329840692803,43329840692803,43329840692803,4811701422724053,52495417243036553

%N The successive approximations up to 5^n for 5-adic integer 2^(1/3).

%C x = ...132203,

%C x^2 = ...344214,

%C x^3 = ...000002 = 2.

%H Seiichi Manyama, <a href="/A290567/b290567.txt">Table of n, a(n) for n = 0..1431</a>

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

%F a(0) = 0 and a(1) = 3, a(n) = a(n-1) + 2 * (a(n-1)^3 - 2) mod 5^n for n > 1.

%e a(1) = ( 3)_5 = 3,

%e a(2) = ( 3)_5 = 3,

%e a(3) = ( 203)_5 = 53,

%e a(4) = (2203)_5 = 303.

%o (PARI) a(n)=truncate((2+O(5^n))^(1/3)); \\ _Joerg Arndt_, Aug 06 2017

%Y Cf. A290566, A290568.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 06 2017