A086237 -id:A086237 - cp's OEIS frontend

A089729 Decimal expansion of Levy's constant 12*log(2)/Pi^2.

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

Offset: 0

Benoit Cloitre, Jan 19 2004

For x>y in [1..n], the average number of loop steps of the Euclid Algorithm for GCD (over all choices x, y) is asymptotic to k*log(n) where k is this constant. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016

			0.8427659132721945169072631939639641155945183893191504965...

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Theorem 2.1.3, p. 84.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Levy Constant

Cf. A086702, A086237.

RealDigits[12 Log[2]/Pi^2, 10, 100][[1]] (* Bruno Berselli, Jun 20 2013 *)

12*log(2)/Pi^2 \\ Michel Marcus, Mar 23 2016

Leading zero removed by R. J. Mathar, Feb 05 2009

A143304 Decimal expansion of Norton's constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 05 2008

The average number of divisions required by the Euclidean algorithm, for a 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 (Norton, 1990). - Amiram Eldar, Aug 27 2020

			0.06535142592303732137...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 157.

Graham H. Norton, On the asymptotic analysis of the Euclidean algorithm, J. Symbolic Comput., Vol. 10 (1990), pp. 53-58.
Eric Weisstein's World of Mathematics, Norton's Constant.

Cf. A001620, A074962, A086237, A306016.

RealDigits[-((Pi^2 - 6*Log[2]*(24 * Log[Glaisher] + 2*EulerGamma + Log[2] - 2 * Log[Pi] - 3))/Pi^2), 10, 100][[1]] (* Amiram Eldar, Aug 27 2020 *)

Equals -((Pi^2 - 6*log(2)*(-3 + 2*EulerGamma + log(2) + 24*log(Glaisher) - 2*log(Pi)))/Pi^2).

Equals (12*log(2)/Pi^2) * (zeta'(2)/zeta(2) - 1/2) + A086237 - 1/2. - Amiram Eldar, Aug 27 2020

A378589 Decimal expansion of (1 - 2*A143304)/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 01 2024

			0.217324287038481339310608686618446034593487731575...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.18, p. 157.

Cf. A143304.

Cf. A001620, A002162, A002388, A053510, A074962, A086237, A225746.

RealDigits[(1+2(Pi^2 - 6*Log[2]*(24*Log[Glaisher] + 2*EulerGamma + Log[2] - 2*Log[Pi] - 3))/Pi^2)/4, 10, 100][[1]]

Equals (1 + 2*(Pi^2 - 6*log(2)*(24*log(A074962) + 2*gamma + log(2) - 2 * log(Pi) - 3))/Pi^2)/4.

cp's OEIS Frontend

A089729 Decimal expansion of Levy's constant 12*log(2)/Pi^2.

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A143304 Decimal expansion of Norton's constant.

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A378589 Decimal expansion of (1 - 2*A143304)/4.

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