A324085 Digits of one of the four 3-adic integers 3^(1/4) that is congruent to 2 mod 13.

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

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 A324153.

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

Table of n, a(n) for n=0..80.

Wikipedia, p-adic number

Formula

Equals A324086*A286838 = A324087*A286839.

a(n) = (A324077(n+1) - A324077(n))/13^n.

For n > 0, a(n) = 12 - A324153(n).

Example

The unique number k in [1, 13^3] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 1211 = (722)_13, so the first three terms are 2, 7 and 7.

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, A324086, A324087, A324153.

Sequence in context: A013671 A019807 A362029 * A340180 A063706 A110779

Adjacent sequences: A324082 A324083 A324084 * A324086 A324087 A324088

Keyword

nonn,base

Author

Jianing Song, Sep 01 2019

Status

approved