A065918 Decimal expansion of log(2 + sqrt(3)).

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, 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, 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

Offset

1,2

Comments

x = 2^n - 1 is prime if and only if x divides cosh(2^(n - 2)*log(2 + sqrt(3))).

Links

Harry J. Smith, Table of n, a(n) for n = 1..2000

Chris Caldwell, Primality Proving, Arndt's theorem.

Index entries for transcendental numbers.

Formula

Equals arccosh(2) since arccosh(x) = log(x + sqrt(x^2 - 1)). - Stanislav Sykora, Nov 01 2013

Equals arctanh(sqrt(3)/2). - Amiram Eldar, Feb 09 2024

Equals log(4) - Sum_{k>=1} (2*k - 1)!!/(k*k!*2^(3*k + 1)). - Antonio Graciá Llorente, Feb 14 2024

Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(3*n - 1/2)*(2*n + 1)). - Antonio Graciá Llorente, Nov 13 2024

Example

1.316957896924816708625046347307968444...

Mathematica

First@ RealDigits[Log[2 + Sqrt@ 3], 10, 102] (* Michael De Vlieger, May 12 2019 *)

Prog

(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
(PARI) acosh(2) \\ Charles R Greathouse IV, Jan 07 2016

Crossrefs

Cf. A001075, A019973, A072877.

Sequence in context: A013610 A008573 A089710 * A362695 A020861 A163213

Adjacent sequences: A065915 A065916 A065917 * A065919 A065920 A065921

Keyword

nonn,easy,cons

Author

Frank Ellermann, Dec 08 2001

Status

approved