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

%I A321690 #14 Nov 21 2018 07:32:40
%S A321690 0,0,0,4,12,28,60,124,124,124,636,1660,1660,1660,9852,9852,9852,9852,
%T A321690 140924,140924,140924,1189500,3286652,7480956,7480956,7480956,7480956,
%U A321690 7480956,7480956,7480956,7480956,7480956,2154964604,2154964604,2154964604,19334833788
%N A321690 Approximations up to 2^n for the 2-adic integer log(5).
%C A321690 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).
%H A321690 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%F A321690 a(n) = Sum_{i=0..n-1} A152228(i)*2^i.
%e A321690 a(3) = (4 + O(2^3)) mod 8 = 4 mod 8 = 4.
%e A321690 a(6) = (4 - 4^2/2 + O(2^6)) mod 64 = (-4) mod 64 = 60.
%e A321690 a(10) = (4 - 4^2/2 + 4^3/3 - 4^4/4 + O(2^10)) mod 1024 = (-140/3) mod 1024 = 636.
%e A321690 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.
%o A321690 (PARI) a(n) = if(n, lift(log(5 + O(2^n))), 0);
%Y A321690 Cf. A321691 (log(-3)).
%K A321690 nonn
%O A321690 0,4
%A A321690 _Jianing Song_, Nov 17 2018