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 A086237 Decimal expansion of Porter's Constant. 3
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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. FORMULA Equals 6*(log(2)/Pi^2)*(3*log(2) + 4*Gamma -(24/Pi^2)*Zeta'(2) - 2) - 1/2. EXAMPLE 1.4670780794339754728977984847072299534499033224149... 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 *) PROG (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 CROSSREFS Cf. A001620, A073002, A143304. Sequence in context: A082237 A042976 A090142 * A200410 A136323 A135798 Adjacent sequences:  A086234 A086235 A086236 * A086238 A086239 A086240 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Jul 12 2003 STATUS approved

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Last modified November 27 09:38 EST 2020. Contains 338679 sequences. (Running on oeis4.)