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A341752
Successive approximations up to 2^n for the 2-adic integer 17^(1/4). This is the 3 (mod 4) case.
3
3, 3, 3, 19, 19, 83, 83, 83, 595, 595, 595, 595, 8787, 8787, 41555, 107091, 107091, 107091, 107091, 107091, 2204243, 6398547, 6398547, 23175763, 56730195, 123839059, 123839059, 123839059, 660709971, 1734451795, 1734451795, 1734451795, 1734451795, 18914320979
OFFSET
2,1
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
FORMULA
a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^4 - 17 is divisible by 2^(n+2), otherwise a(n-1) + 2^(n-1).
a(n) = 2^n - A341751(n).
a(n) = Sum_{i=0..n-1} A341754(i)*2^i.
a(n)^2 == A341538(n) (mod 2^n).
EXAMPLE
The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^4 - 17 is divisible by 16 is 3, so a(2) = 3.
a(2)^4 - 17 = 64 which is divisible by 32, so a(3) = a(2) = 3.
a(3)^4 - 17 = 64 which is divisible by 64, so a(4) = a(3) = 3.
a(4)^4 - 17 = 64 which is not divisible by 128, so a(5) = a(4) + 2^4 = 19.
a(5)^4 - 17 = 130304 which is ndivisible by 256, so a(6) = a(5) = 19.
...
PROG
(PARI) a(n) = truncate(-sqrtn(17+O(2^(n+2)), 4))
CROSSREFS
Cf. A341754 (digits of the associated 2-adic fourth root of 17), A341538.
Approximations of p-adic fourth-power roots:
A341751, this sequence (2-adic, 17^(1/4));
A325484, A325485, A325486, A325487 (5-adic, 6^(1/4));
A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).
Sequence in context: A342363 A229934 A239125 * A325892 A127014 A073748
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 18 2021
STATUS
approved