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%I #45 Feb 17 2024 12:42:10
%S 1,3,1,6,9,5,7,8,9,6,9,2,4,8,1,6,7,0,8,6,2,5,0,4,6,3,4,7,3,0,7,9,6,8,
%T 4,4,4,0,2,6,9,8,1,9,7,1,4,6,7,5,1,6,4,7,9,7,6,8,4,7,2,2,5,6,9,2,0,4,
%U 6,0,1,8,5,4,1,6,4,4,3,9,7,6,0,7,4,2,1,9,0,1,3,4,5,0,1,0,1,7,8,3,5,5
%N Decimal expansion of log(2 + sqrt(3)).
%C x = 2^n - 1 is prime if and only if x divides cosh(2^(n - 2)*log(2 + sqrt(3))).
%H Harry J. Smith, <a href="/A065918/b065918.txt">Table of n, a(n) for n = 1..2000</a>
%H Chris Caldwell, <a href="http://www.utm.edu/research/primes/prove/prove3_2.html">Primality Proving</a>, Arndt's theorem.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals arccosh(2) since arccosh(x) = log(x + sqrt(x^2 - 1)). - _Stanislav Sykora_, Nov 01 2013
%F Equals arctanh(sqrt(3)/2). - _Amiram Eldar_, Feb 09 2024
%F Equals log(4) - Sum_{k>=1} (2*k - 1)!!/(k*k!*2^(3*k + 1)). - _Antonio Graciá Llorente_, Feb 14 2024
%e 1.316957896924816708625046347307968444...
%t First@ RealDigits[Log[2 + Sqrt@ 3], 10, 102] (* _Michael De Vlieger_, May 12 2019 *)
%o (PARI) default(realprecision, 2080); x=log(2 + sqrt(3)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065918.txt", n, " ", d)) \\ _Harry J. Smith_, Nov 04 2009
%o (PARI) acosh(2) \\ _Charles R Greathouse IV_, Jan 07 2016
%Y Cf. A001075, A019973, A072877.
%K nonn,easy,cons
%O 1,2
%A _Frank Ellermann_, Dec 08 2001
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