A212152 - OEIS

%I #22 Aug 26 2022 23:41:01

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

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

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

%N Digits of one of the three 7-adic integers (-1)^(1/3).

%C See A210852 for comments and an approximation to this 7-adic number, called there u.  See also A048898 for references on p-adic numbers.

%C a(n), n>=1, is the (unique) solution of the linear congruence 3 * b(n)^2 * a(n) + c(n) == 0 (mod 7), with  b(n):=A210852(n) and c(n):=A210853(n). a(0) = 3, one of the three solutions of x^3+1 == 0 (mod 7).

%C Since b(n) == 3 (mod 7), a(n) == c(n) (mod 7) for n>0. - _Álvar Ibeas_, Feb 20 2017

%C With a(0) = 2, this is the digits of one of the three cube root of 1, the one that is congruent to 2 modulo 7. - _Jianing Song_, Aug 26 2022

%H Robert Israel, <a href="/A212152/b212152.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = (b(n+1) - b(n))/7^n, n>=1, with b(n):=A210852(n), defined by a recurrence given there. One also finds a Maple program for b(n) there. a(0)=3.

%p op([1,1,3],select(t -> padic:-ratvaluep(t,1)=3, [padic:-rootp(x^3+1,7,100)])); # _Robert Israel_, Mar 27 2018

%Y Cf. A210852 (approximations of (-1)^(1/3)), A212155 (digits of another cube root of -1), 6*A000012 (digits of -1).

%Y Cf. A210850, A210851 (digits of the 5-adic integers sqrt(-1)); A319297, A319305, A319555 (digits of the 7-adic integers 6^(1/3)).

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, May 02 2012