A321690 Approximations up to 2^n for the 2-adic integer log(5).

0, 0, 0, 4, 12, 28, 60, 124, 124, 124, 636, 1660, 1660, 1660, 9852, 9852, 9852, 9852, 140924, 140924, 140924, 1189500, 3286652, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 2154964604, 2154964604, 2154964604, 19334833788

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 = (-4) mod 64 = 60.
a(10) = (4 - 4^2/2 + 4^3/3 - 4^4/4 + O(2^10)) mod 1024 = (-140/3) mod 1024 = 636.
a(11) = (4 - 4^2/2 + 4^3/3 - 4^4/4 + 4^5/5 + O(2^11)) mod 2048 = (2372/15) mod 2048 = 1660.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A321691 (log(-3)).

Programs

PARI

a(n) = if(n, lift(log(5 + O(2^n))), 0);

Formula

a(n) = Sum_{i=0..n-1} A152228(i)*2^i.