A341751 Successive approximations up to 2^n for the 2-adic integer 17^(1/4). This is the 1 (mod 4) case.

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

Jianing Song, Table of n, a(n) for n = 2..1000

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

Adjacent sequences: A341748 A341749 A341750 * A341752 A341753 A341754

Keyword

nonn

Author

Jianing Song, Feb 18 2021

Status

approved