A053510 - OEIS

%I #50 Mar 27 2024 20:11:54

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

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

%U 3,7,7,6,9,8,8,4,8,2,6,0,7,9,7,8,3,6,2,3,2,7,0,2,7,5,4,8,9,7,0,7,7,0,2,0,0,9

%N Decimal expansion of log(Pi).

%C Also the least positive x such that sin(exp(x))==0.

%C Also real part of log(log(-1)). - _Stanislav Sykora_, May 11 2015

%C Cheng, Dietel, Herblot, Huang, Krieger, Marques, Mason, Mereb, & Wilson show, expanding a remark by S. Lang, that Schanuel's conjecture implies that this constant and Pi are algebraically independent over a set E which includes the algebraic numbers and (in a technical sense) allows any finite number of exponentiations, see the paper for details and a still more general result. - _Charles R Greathouse IV_, Dec 15 2019

%D Wolfram Research, 1991 Mathematica Conference, Elementary Tutorial Notes, Section 1, Introduction to Mathematica, Paul Abbott, page 25.

%H G. C. Greubel, <a href="/A053510/b053510.txt">Table of n, a(n) for n = 1..5000</a>

%H Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, S. Robert Wilson, <a href="https://doi.org/10.1016/j.jnt.2008.10.018">Some consequences of Schanuel's conjecture</a>, Journal of Number Theory 129:6 (2009), pp. 1464-1467.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=OcBLfMJyd0k">Frullani Integral</a>, YouTube video, 2021.

%F Equals log(log(-1)) - (Pi/2)*I. - _Stanislav Sykora_, May 11 2015

%F Equals 1 + Sum_{n>=1} zeta(2*n)/(n*(2*n+1)*2^(2*n)), where zeta is the Riemann zeta function. - _Vaclav Kotesovec_, Mar 04 2016

%F Equals 3/2 - Sum_{k>=1} (zeta(2*k)-1)/(k+1). - _Vaclav Kotesovec_, Jun 19 2021

%e 1.1447298858494001741...

%t RealDigits[Log[Pi], 10, 111][[1]]

%o (PARI) log(Pi) \\ _Charles R Greathouse IV_, Jan 04 2016

%o (Magma) R:= RealField(100); Log(Pi(R)); // _G. C. Greubel_, May 15 2019

%o (SageMath) numerical_approx(log(pi), digits=100) # _G. C. Greubel_, May 15 2019

%Y Cf. A000796, A053511.

%K cons,easy,nonn

%O 1,3

%A Hsu, Po-Wei (Benny) (arsene_lupin(AT)intekom.co.za), Jan 14 2000

%E More terms from _James A. Sellers_, Jan 20 2000