A322934 The successive approximations up to 2^n for 2-adic integer 7^(1/3).

0, 1, 3, 7, 7, 23, 23, 23, 151, 407, 407, 1431, 3479, 3479, 11671, 11671, 44439, 109975, 241047, 503191, 1027479, 2076055, 2076055, 6270359, 6270359, 6270359, 6270359, 6270359, 6270359, 274705815, 811576727, 1885318551, 1885318551, 6180285847

Offset: 0

Author

Jianing Song, Aug 30 2019

Keywords

• nonn

Comments

a(n) is the unique solution to x^3 == 7 (mod 2^n) in the range [0, 2^n - 1].

Examples

			7^3 = 343 = 21*2^4 + 7;
23^3 = 12167 = 380*2^5 + 7 = 190*2^6 + 7 = 95*2^7 + 7;
151^3 = 3442951 = 13449*2^8 + 7.
		

Links

• Wikipedia, p-adic number

Crossrefs

For the digits of 7^(1/3), see A323095.

Approximations of p-adic cubic roots:

A322701 (2-adic, 3^(1/3));

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

this sequence (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)).

Programs

PARI

a(n) = lift(sqrtn(7+O(2^n), 3))

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^3 - 7 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).