A321108 Digits of one of the three 13-adic integers 5^(1/3) that is related to A321105.

11, 11, 5, 11, 2, 0, 9, 0, 6, 11, 9, 6, 7, 9, 2, 9, 9, 2, 3, 3, 8, 2, 7, 11, 6, 7, 4, 7, 10, 5, 5, 4, 11, 6, 2, 5, 2, 7, 10, 9, 9, 2, 9, 5, 7, 7, 4, 5, 10, 4, 1, 6, 4, 1, 4, 0, 4, 10, 11, 4, 12, 12, 7, 2, 9, 6, 11, 8, 5, 6, 11, 2, 0, 6, 6, 12, 10, 8, 12, 11, 2

Offset

0,1

Comments

For k not divisible by 5, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Wikipedia, p-adic number

Formula

a(n) = (A321105(n+1) - A321105(n))/13^n.

Example

The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 999 = (5BB)_13, so the first three terms are 11, 11 and 5.

Prog

(PARI) a(n) = lift(sqrtn(5+O(13^(n+1)), 3) * (-1-sqrt(-3+O(13^(n+1))))/2)\13^n

Crossrefs

Cf. A320914, A320915, A321105, A321106, A321107.

For 5-adic cubic roots, see A290566, A290563, A309443.

Sequence in context: A135684 A220295 A300289 * A126610 A087380 A152986

Adjacent sequences: A321105 A321106 A321107 * A321109 A321110 A321111

Keyword

nonn,base

Author

Jianing Song, Aug 27 2019

Status

approved