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%I #27 Nov 18 2018 10:01:55
%S 0,1,3,3,3,3,35,35,163,163,675,1699,1699,1699,9891,9891,42659,42659,
%T 42659,304803,304803,304803,304803,4499107,4499107,21276323,21276323,
%U 21276323,155494051,423929507,423929507,1497671331,1497671331,1497671331,10087605923
%N Approximations up to 2^n for 2-adic integer log_5(-3).
%C a(n) is the unique number x in [0, 2^(n-2) - 1] such that 5^x == -3 (mod 2^n). This is well defined because {5^x mod 2^n : 0 <= x <= 2^(n-2) - 1} = {1, 5, 9, ..., 2^n - 3}.
%C For any odd 2-adic integer x, define log(x) = -Sum_{k>=1} (1 - x)^k/k (which always converges over the 2-adic field) and log_x(y) = log(y)/log(x), then we have log(-1) = 0. If we further define exp(x) = Sum_{k>=0} x^k/k! for 2-adic integers divisible by 4, then we have exp(log(x)) = x if and only if x == 1 (mod 4). As a result, log_5(-3) = log_5(3) = log_(-5)(3) = log_(-5)(-3), but it's better to be stated as log_5(-3).
%C For n > 0, a(n) is also the unique number x in [0, 2^(n-2) - 1] such that (-5)^x == 3 (mod 2^n).
%C a(n) is the multiplicative inverse of A321082(n) modulo 2^(n-2).
%H Jianing Song, <a href="/A321080/b321080.txt">Table of n, a(n) for n = 2..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%F a(2) = 0; for n >= 3, a(n) = a(n-1) if 5^a(n-1) + 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-3).
%F a(n) = Sum_{i=0..n-3} A321081(i)*2^i (empty sum yields 0 for n = 2).
%F a(n) = A321691(n+2)/A321690(n+2) mod 2^n.
%e The only number in the range [0, 2^(n-2) - 1] for n = 2 is 0, so a(2) = 0.
%e 5^a(2) + 3 = 4 which is not divisible by 8, so a(3) = a(2) + 2^0 = 1.
%e 5^a(3) + 3 = 8 which is not divisible by 16, so a(4) = a(3) + 2^1 = 3.
%e 5^a(4) + 3 = 128 which is divisible by 32, 64 and 128 but not 256, so a(5) = a(6) = a(7) = a(4) = 3, a(8) = a(7) + 2^5 = 35.
%e 5^a(8) + 3 = ... which is divisible by 512 but not 1024, so a(9) = a(8) = 35, a(10) = a(9) + 2^7 = 163.
%o (PARI) b(n) = {my(v=vector(n)); for(n=3, n, v[n] = v[n-1] + if(Mod(5,2^n)^v[n-1] + 3==0, 0, 2^(n-3))); v}
%o a(n) = b(n)[n] \\ Program provided by _Andrew Howroyd_
%o (PARI) a(n)={if(n<3, 0, truncate(log(-3 + O(2^n))/log(5 + O(2^n))))} \\ _Andrew Howroyd_, Nov 03 2018
%Y Cf. A321081, A321082, A321690, A321691.
%K nonn
%O 2,3
%A _Jianing Song_, Oct 27 2018
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