OFFSET
2,2
COMMENTS
a(n) is the unique number k in [1, 2^n] and congruent to 1 mod 4 such that k^4 - 17 is divisible by 2^(n+2).
For odd k, k has a fourth root in the ring of 2-adic integers if and only if k == 1 (mod 16), in which case k has exactly two fourth roots.
LINKS
Jianing Song, Table of n, a(n) for n = 2..1000
FORMULA
EXAMPLE
The unique number k in [1, 4] and congruent to 1 modulo 4 such that k^4 - 17 is divisible by 16 is 1, so a(2) = 1.
a(2)^4 - 17 = -16 which is not divisible by 32, so a(3) = a(2) + 2^2 = 5.
a(3)^4 - 17 = 608 which is not divisible by 64, so a(4) = a(3) + 2^3 = 13.
a(4)^4 - 17 = 28544 which is divisible by 128, so a(5) = a(4) = 13.
a(5)^4 - 17 = 28544 which is not ndivisible by 256, so a(6) = a(5) + 2^5 = 45.
...
PROG
(PARI) a(n) = truncate(sqrtn(17+O(2^(n+2)), 4))
CROSSREFS
Approximations of p-adic fourth-power roots:
this sequence, A341752 (2-adic, 17^(1/4));
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 18 2021
STATUS
approved