OFFSET
0,2
COMMENTS
For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 7 mod 13 such that k^3 - 5 is divisible by 13^n.
For k not divisible by 13, 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
Wikipedia, p-adic number
EXAMPLE
The unique number k in [1, 13^2] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 7, so a(2) = 7.
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, so a(3) = 1021.
PROG
(PARI) a(n) = lift(sqrtn(5+O(13^n), 3) * (-1+sqrt(-3+O(13^n)))/2)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 27 2019
STATUS
approved