A324153 Digits of one of the four 13-adic integers 3^(1/4) that is congruent to 11 mod 13.

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

Offset

0,1

Comments

One of the two square roots of A322087, where an A-number represents a 13-adic number. The other square root is A324085.

For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-adic field, then k has exactly 4 fourth-power roots.

Links

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

Wikipedia, p-adic number

Formula

Equals A324086*A286839 = A324087*A286838.

a(n) = (A324084(n+1) - A324084(n))/13^n.

For n > 0, a(n) = 12 - A324085(n).

Example

The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 986 = (5AB)_13, so the first three terms are 11, 10 and 5.

Maple

R:= select(t -> op([1, 3, 1], t)=11, [padic:-rootp(x^4-3, 13, 101)]):
op([1, 1, 3], R); # Robert Israel, Sep 08 2019

Prog

(PARI) a(n) = lift(-sqrtn(3+O(13^(n+1)), 4) * sqrt(-1+O(13^(n+1))))\13^n

Crossrefs

Cf. A286838, A286839, A322087, A324077, A324082, A324083, A324084, A324085, A324086, A324087.

Sequence in context: A256078 A078200 A105034 * A065001 A022967 A023453

Adjacent sequences: A324150 A324151 A324152 * A324154 A324155 A324156

Keyword

nonn,base

Author

Jianing Song, Sep 01 2019

Status

approved