|
%I #44 Jul 29 2021 04:25:31
%S 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,
%T 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,
%U 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
%N Decimal expansion of Porter's Constant.
%C 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
%C 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
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, p. 157
%D Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 113.
%H G. C. Greubel, <a href="/A086237/b086237.txt">Table of n, a(n) for n = 1..10000</a>
%H Donald E. Knuth, <a href="http://dx.doi.org/10.1016/0898-1221(76)90025-0">Evaluation of Porter's constant</a>, Computers and Mathematics with Applications, Vol. 2, No. 2 (1976), pp. 137-139.
%H Gustav Lochs, <a href="https://doi.org/10.1007/BF01322657">Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmäßigen Kettenbrüche</a>, Monatshefte für Mathematik, Vol. 65, No. 1 (1961), pp. 27-52, <a href="https://eudml.org/doc/177113">alternative link</a>.
%H John William Porter, <a href="http://dx.doi.org/10.1112/S0025579300004459">On a Theorem of Heilbronn</a>, Mathematika, Vol. 22, No. 1 (1975), pp. 20-28.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PortersConstant.html">Porter's Constant</a>.
%F Equals 6*(log(2)/Pi^2)*(3*log(2) + 4*Gamma -(24/Pi^2)*Zeta'(2) - 2) - 1/2.
%e 1.4670780794339754728977984847072299534499033224148...
%t 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 *)
%t 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 *)
%o (PARI) x=.25^default(realprecision)
%o (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
%o (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
%o (PARI) 6*log(2)/Pi^2*(3*log(2) + 4*Euler - 24/Pi^2*zeta'(2) - 2) - 1/2 \\ _Michel Marcus_, Aug 27 2014
%Y Cf. A001620, A073002, A143304.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 12 2003
|
