A322701 The successive approximations up to 2^n for 2-adic integer 3^(1/3).

0, 1, 3, 3, 11, 27, 59, 123, 123, 379, 379, 379, 379, 4475, 12667, 29051, 61819, 127355, 127355, 127355, 127355, 127355, 2224507, 2224507, 2224507, 19001723, 52556155, 119665019, 253882747, 253882747, 253882747, 1327624571, 3475108219, 7770075515

Offset

0,3

Comments

a(n) is the unique solution to x^3 == 3 (mod 2^n) in the range [0, 2^n - 1].

Links

Table of n, a(n) for n=0..33.

Wikipedia, p-adic number

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^3 - 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

Example

11^3 = 1331 = 83*2^4 + 3;
27^3 = 19683 = 615*2^5 + 3;
59^3 = 205379 = 3209*2^6 + 3.

Prog

(PARI) a(n) = lift(sqrtn(3+O(2^n), 3))

Crossrefs

For the digits of 3^(1/3), see A323000.

Approximations of p-adic cubic roots:

this sequence (2-adic, 3^(1/3));

A322926 (2-adic, 5^(1/3));

A322934 (2-adic, 7^(1/3));

A322999 (2-adic, 9^(1/3));

A290567 (5-adic, 2^(1/3));

A290568 (5-adic, 3^(1/3));

A309444 (5-adic, 4^(1/3));

A319097, A319098, A319199 (7-adic, 6^(1/3));

A320914, A320915, A321105 (13-adic, 5^(1/3)).

Sequence in context: A281639 A281101 A341601 * A124265 A163938 A373393

Adjacent sequences: A322698 A322699 A322700 * A322702 A322703 A322704

Keyword

nonn

Author

Jianing Song, Aug 30 2019

Status

approved