

A324083


One of the four successive approximations up to 13^n for 13adic integer 3^(1/4).This is the 10 (mod 13) case (except for n = 0).


13



0, 10, 101, 1622, 14804, 214731, 214731, 9868349, 637353519, 637353519, 637353519, 552071320915, 552071320915, 23850156443396, 1538225689404661, 48786742317796129, 560645672458703699, 5218561936740962586, 13868977856122300519, 126324384808079693648
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OFFSET

0,2


COMMENTS

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 10 mod 13 such that k^4  3 is divisible by 13^n.
For k not divisible by 13, k is a fourth power in 13adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13adic field, then k has exactly 4 fourthpower roots.


LINKS

Table of n, a(n) for n=0..19.
Wikipedia, padic number


FORMULA

a(n) = A324077(n)*A286840(n) mod 13^n = A324084(n)*A286841(n) mod 13^n.
For n > 0, a(n) = 13^n  A324082(n).
a(n)^2 == A322086(n) (mod 13^n).


EXAMPLE

The unique number k in [1, 13^2] and congruent to 10 modulo 13 such that k^4  3 is divisible by 13^2 is k = 101, so a(2) = 101.
The unique number k in [1, 13^3] and congruent to 10 modulo 13 such that k^4  3 is divisible by 13^3 is k = 1622, so a(3) = 1622.


PROG

(PARI) a(n) = lift(sqrtn(3+O(13^n), 4))


CROSSREFS

Cf. A286840, A286841, A322085, A324077, A324082, A324084, A324085, A324086, A324087, A324153.
Sequence in context: A287014 A108892 A309802 * A162849 A041182 A036299
Adjacent sequences: A324080 A324081 A324082 * A324084 A324085 A324086


KEYWORD

nonn


AUTHOR

Jianing Song, Sep 01 2019


STATUS

approved



