A321691 Approximations up to 2^n for the 2-adic integer log(-3).

0, 0, 0, 4, 4, 20, 52, 116, 244, 244, 244, 244, 2292, 2292, 10484, 26868, 59636, 59636, 190708, 190708, 190708, 1239284, 1239284, 1239284, 9627892, 9627892, 43182324, 43182324, 43182324, 43182324, 580053236, 580053236, 580053236, 4875020532

Offset: 0

Author

Jianing Song, Nov 17 2018

Keywords

• nonn

Comments

Let 4Q_2 = {x belongs to Q_2 : |x|2 <= 1/4} and 4Q_2 + 1 = {x belongs to Q_2: |x - 1|_2 <= 1/4}. Define exp(x) = Sum{k>=0} x^k/k! and log(x) = -Sum_{k>=1} (1 - x)^k/k over 2-adic field, then exp(x) is a one-to-one mapping from 4Q_2 to 4Q_2 + 1, and log(x) is the inverse of exp(x).

Examples

			a(3) = (-4 + O(2^3)) mod 8 = (-4) mod 8 = 4.
a(6) = (-4 - 4^2/2 - O(2^6)) mod 64 = (-12) mod 64 = 52.
a(10) = (-4 - 4^2/2 - 4^3/3 - 4^4/4 - O(2^10)) mod 1024 = (-292/3) mod 1024 = 244.
a(11) = (-4 - 4^2/2 - 4^3/3 - 4^4/4 - 4^5/5 - O(2^11)) mod 2048 = (-4532/15) mod 2048 = 244.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A321690 (log(5)), A321694.

Programs

PARI

a(n) = if(n, lift(log(-3 + O(2^n))), 0);

Formula

a(n) = Sum_{i=0..n-1} A321694(i)*2^i.

Conjecture: a(n) = 2*A309753(n-1). - R. J. Mathar, Aug 06 2023