A053510 Decimal expansion of log(Pi).

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, 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, 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

Offset

1,3

Comments

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

Also real part of log(log(-1)). - Stanislav Sykora, May 11 2015

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

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, S. Robert Wilson, Some consequences of Schanuel's conjecture, Journal of Number Theory 129:6 (2009), pp. 1464-1467.

Michael Penn, Frullani Integral, YouTube video, 2021.

Formula

Equals log(log(-1)) - (Pi/2)*I. - Stanislav Sykora, May 11 2015

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

Equals 3/2 - Sum_{k>=1} (zeta(2*k)-1)/(k+1). - Vaclav Kotesovec, Jun 19 2021

Example

1.1447298858494001741...

Mathematica

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

Prog

(PARI) log(Pi) \\ Charles R Greathouse IV, Jan 04 2016
(Magma) R:= RealField(100); Log(Pi(R)); // G. C. Greubel, May 15 2019
(SageMath) numerical_approx(log(pi), digits=100) # G. C. Greubel, May 15 2019

Crossrefs

Cf. A000796, A053511.

Sequence in context: A010476 A020762 A204158 * A197009 A204156 A163106

Adjacent sequences: A053507 A053508 A053509 * A053511 A053512 A053513

Keyword

cons,easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jan 20 2000

Status

approved