A321107 Digits of one of the three 13-adic integers 5^(1/3) that is related to A320915.

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

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) = (A320915(n+1) - A320915(n))/13^n.

Example

The unique number k in [1, 13^3] and congruent to 8 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 177 = (108)_13, so the first three terms are 8, 0 and 1.

Prog

(PARI) a(n) = lift(sqrtn(5+O(13^(n+1)), 3))\13^n

Crossrefs

Cf. A320914, A320915, A321105, A321106, A321108.

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

Sequence in context: A011470 A345295 A198940 * A198117 A241215 A272343

Adjacent sequences: A321104 A321105 A321106 * A321108 A321109 A321110

Keyword

nonn,base

Author

Jianing Song, Aug 27 2019

Status

approved