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

Author

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

Keywords

• cons
• easy
• nonn

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

Examples

			1.1447298858494001741...
		

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.

Crossrefs

Cf. A000796, A053511.

Programs

Magma

R:= RealField(100); Log(Pi(R)); // G. C. Greubel, May 15 2019

Mathematica

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

PARI

log(Pi) \\ Charles R Greathouse IV, Jan 04 2016

SageMath

numerical_approx(log(pi), digits=100) # G. C. Greubel, May 15 2019

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

Extensions

More terms from James Sellers, Jan 20 2000