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A324085
Digits of one of the four 3-adic integers 3^(1/4) that is congruent to 2 mod 13.
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.
FORMULA
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
KEYWORD
nonn,base
AUTHOR
Jianing Song, Sep 01 2019
STATUS
approved