A086237 Decimal expansion of Porter's constant.

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

Offset: 1

Author

Eric W. Weisstein, Jul 12 2003

Keywords

• nonn
• cons

Comments

In his 'Addendum' to his paper in the year 2000 Don Knuth writes: "Gustav Lochs deserves to be mentioned here, because his work preceded that of Porter by nearly 15 years and involved essentially the same constant. Perhaps we should [..] refer in future to the Lochs-Porter constant, instead of simply saying 'Porter's constant'." - Peter Luschny, Aug 26 2014

The average number of divisions required by the Euclidean algorithm, for a coprime pair of independently and uniformly chosen numbers in the range [1, N] is (12*log(2)/Pi^2) * log(N) + c + O(N^(e-1/6)), for any e>0, where c is this constant (Porter, 1975). - Amiram Eldar, Aug 27 2020

Examples

			1.4670780794339754728977984847072299534499033224148...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, p. 157
• Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 113.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Donald E. Knuth, Evaluation of Porter's constant, Computers and Mathematics with Applications, Vol. 2, No. 2 (1976), pp. 137-139.
• Gustav Lochs, Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmäßigen Kettenbrüche, Monatshefte für Mathematik, Vol. 65, No. 1 (1961), pp. 27-52, alternative link.
• John William Porter, On a Theorem of Heilbronn, Mathematika, Vol. 22, No. 1 (1975), pp. 20-28.
• Eric Weisstein's World of Mathematics, Porter's Constant.

Crossrefs

Cf. A001620, A073002, A074962, A143304.

Programs

Mathematica

RealDigits[(6 Log[2] (48 Log[Glaisher] - Log[2] - 4 Log[Pi] - 2))/Pi^2 - 1/2, 10, 110][[1]] (* Eric W. Weisstein, Aug 22 2013 *)
RealDigits[(6 Log[2] (Pi^2 (-2 + 4 EulerGamma + Log[8]) - 24 Zeta'[2]))/Pi^4 - 1/2, 10, 110][[1]] (* Eric W. Weisstein, Aug 22 2013 *)

PARI

x=.25^default(realprecision)
(6*log(2)*(4-48*(zeta(-1+x)-zeta(-1))/x-log(2)-4*log(Pi)-2))/Pi^2 - 1/2 \\ Charles R Greathouse IV, Aug 22 2013

PARI

(6*log(2)*(4-48*zeta'(-1)-log(2)-4*log(Pi)-2))/Pi^2-1/2 \\ Charles R Greathouse IV, Dec 12 2013

PARI

6*log(2)/Pi^2*(3*log(2) + 4*Euler - 24/Pi^2*zeta'(2) - 2) - 1/2 \\ Michel Marcus, Aug 27 2014

Formula

Equals 6*(log(2)/Pi^2)*(3*log(2) + 4*Gamma -(24/Pi^2)*Zeta'(2) - 2) - 1/2.

Equals 6*log(2)*(48*log(A074962) - 4*log(Pi) - log(2) - 2)/Pi^2 - 1/2 (see Finch). - Stefano Spezia, Dec 01 2024