OFFSET
0,5
COMMENTS
a(n) is the unique solution to x^5 == 9 (mod 2^n) in the range [0, 2^n - 1].
LINKS
Wikipedia, p-adic number
FORMULA
For n > 0, a(n) = a(n-1) if a(n-1)^5 - 9 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
EXAMPLE
For n = 2, the unique solution to x^5 == 9 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1.
a(2)^5 - 9 = -8 which is divisible by 8, so a(3) = a(2) = 1;
a(3)^5 - 9 = -8 which is not divisible by 16, so a(4) = a(3) + 8 = 9;
a(4)^5 - 9 = 59040 which is divisible by 32, so a(5) = a(4) = 9;
a(5)^5 - 9 = 59040 which is not divisible by 64, so a(6) = a(5) + 32 = 41.
PROG
(PARI) a(n) = lift(sqrtn(9+O(2^n), 5))
CROSSREFS
For the digits of 9^(1/5), see A325899.
Approximations of p-adic fifth-power roots:
A325892 (2-adic, 3^(1/5));
A325893 (2-adic, 5^(1/5));
A325894 (2-adic, 7^(1/5));
this sequence (2-adic, 9^(1/5));
A322157 (5-adic, 7^(1/5));
A309450 (7-adic, 2^(1/5));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 07 2019
STATUS
approved