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Decimal expansion of sqrt(log 2).
1

%I #24 Sep 01 2021 12:26:36

%S 8,3,2,5,5,4,6,1,1,1,5,7,6,9,7,7,5,6,3,5,3,1,6,4,6,4,4,8,9,5,2,0,1,0,

%T 4,7,6,3,0,5,8,8,8,5,2,2,6,4,4,4,0,7,2,9,1,6,6,8,2,9,1,1,7,2,3,4,0,7,

%U 9,4,3,5,1,9,7,3,0,4,6,3,7,1,4,8,9,9,8,0

%N Decimal expansion of sqrt(log 2).

%C Represents a transcendental number.

%D Ludwig Seidel, Ueber eine Darstellung des Kreisbogens, des Logarithmus und des elliptischen Integrales erster Art durch unendliche Producte, Borchardt J., (1871), vol. 73, pp. 273-291.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Product_{k>=1} (2/(2^(1/2^k) + 1))^(1/2).

%F Equals sqrt(2*arccoth(3)) = sqrt(A002162).

%e 0.8325546111576977563531646448952010476305888522644407291668291172340794351973...

%p Digits := 120; sqrt(log(2)): evalf(%)*10^91:

%p ListTools:-Reverse(convert(floor(%), base, 10));

%t RealDigits[Sqrt[Log[2]], 10, 100][[1]] (* _Amiram Eldar_, Sep 01 2021 *)

%o (Julia)

%o using Nemo

%o R = RealField(305); _1 = R(1); _2 = R(2); H = R(1/2)

%o p = prod((_2/(_2^(_1/_2^k) + 1))^H for k in 1:300)

%o println(p)

%Y Cf. A002162.

%K nonn,cons

%O 0,1

%A _Peter Luschny_, Sep 01 2021