A212152 Digits of one of the three 7-adic integers (-1)^(1/3).

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, 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, 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

Offset: 0

Author

Wolfdieter Lang, May 02 2012

Keywords

• nonn
• easy

Comments

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

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).

Since b(n) == 3 (mod 7), a(n) == c(n) (mod 7) for n>0. - Álvar Ibeas, Feb 20 2017

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

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

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

Programs

Maple

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

Mathematica

Join[{3}, MapIndexed[#/7^#2[[1]] &, Differences[FoldList[PowerMod[#, 7, 7^#2] &, 3, Range[2, 100]]]]] (* Paolo Xausa, Jan 14 2025 *)

Formula

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.