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

0,4

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).

Links

Table of n, a(n) for n=0..35.

Wikipedia, p-adic number

Formula

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

Example

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.

Prog

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

Crossrefs

Cf. A321691 (log(-3)).

Sequence in context: A186924 A261320 A179023 * A269712 A028399 A173033

Adjacent sequences: A321687 A321688 A321689 * A321691 A321692 A321693

Keyword

nonn

Author

Jianing Song, Nov 17 2018

Status

approved