A321106 - OEIS

A321106

Digits of one of the three 13-adic integers 5^(1/3) that is related to A320914.

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

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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) = (A320914(n+1) - A320914(n))/13^n.

EXAMPLE

The unique number k in [1, 13^3] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 1021 = (607)_13, so the first three terms are 7, 0 and 6.

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, A321107, A321108.

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

Sequence in context: A021938 A076421 A196763 * A225457 A188928 A036479

Adjacent sequences: A321103 A321104 A321105 * A321107 A321108 A321109

KEYWORD

nonn,base

AUTHOR

Jianing Song, Aug 27 2019

STATUS

approved