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A341751
Successive approximations up to 2^n for the 2-adic integer 17^(1/4). This is the 1 (mod 4) case.
3
1, 5, 13, 13, 45, 45, 173, 429, 429, 1453, 3501, 7597, 7597, 23981, 23981, 23981, 155053, 417197, 941485, 1990061, 1990061, 1990061, 10378669, 10378669, 10378669, 10378669, 144596397, 413031853, 413031853, 413031853, 2560515501, 6855482797, 15445417389, 15445417389
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
FORMULA
a(2) = 1; 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 - A341752(n).
a(n) = Sum_{i=0..n-1} A341753(i)*2^i.
a(n)^2 == A341538(n) (mod 2^n).
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
Cf. A341753 (digits of the associated 2-adic fourth root of 17), A341538.
Approximations of p-adic fourth-power roots:
this sequence, A341752 (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: A321992 A231806 A183782 * A244435 A134202 A309621
KEYWORD
nonn
AUTHOR
Jianing Song, Feb 18 2021
STATUS
approved