A242673 -id:A242673 - cp's OEIS frontend

A242672 Decimal expansion of an optimal stopping constant related to the Secretary problem.

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

Offset: 1

Jean-François Alcover, May 20 2014

			3.869519241397999495694167278779...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

Eric Weisstein's MathWorld, Sultan's Dowry Problem.
Wikipedia, Secretary problem.

Cf. A054404, A242673, A242674.

NProduct[(1+2/k)^(1/(k+1)), {k, 1, Infinity}, NProductFactors -> 1000, WorkingPrecision -> 48] // RealDigits[#, 10, 40]& // First

product_(k>0) (1+2/k)^(1/(k+1)).

Also equals exp(2*Sum_{k>=3} (log(k)/(k^2-1)) - log(2)/3).

More terms from Alois P. Heinz, Jun 06 2014

A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

This is the asymptotic probability of success for the full-information problem with uniform distribution: we can not only determine which of any two applicants is better than the other, but also determine his/her absolute value, and that value is known to be uniformly distributed on a known interval (say, [0, 1]), independently for each applicant; so we have more information than in the basic version of the problem (for which the chance of success is given by A068985), so the chance of success is greater. Here the number of applicants is known in advance (although we consider the limiting case when it is sent to infinity); for the variant where it is itself a random variable, see A325905. - Andrey Zabolotskiy, Sep 14 2019

			0.580164223920855346426008323572997276633...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A054404, A242672, A242673, A068985, A325905.

a = x /. FindRoot[ExpIntegralEi[x] - EulerGamma - Log[x] == 1, {x, 1}, WorkingPrecision -> 105]; Exp[-a] - (Exp[a]-a-1)*ExpIntegralEi[-a] // RealDigits[#, 10, 100]& // First

exp(-a) - (exp(a)-a-1)*Ei(-a), where a is the unique real solution of the equation Ei(a)-gamma-log(a) = 1, Ei being the exponential integral function, and gamma the Euler-Mascheroni constant (0.5772156649...).

A245771 Decimal expansion of 'b', an optimal stopping constant associated with the secretary problem when the objective is to maximize the hiree's expected quality.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 01 2014

			1.767993786136154050443634406781132331077681433131956515576986059626...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

Steven Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024.
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 9.
Steven Finch, Half-Iterates of x(1+x), sin(x) and exp(x/e), arXiv:2506.07625 [math.NT], 2025. See p. 4.
Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.
Eric Weisstein's MathWorld, Sultan's Dowry Problem.
Wikipedia, Secretary problem.

Cf. A054404, A242672, A242673, A242674, A243533.

// See the link to Jon E. Schoenfield's program.

nmax = 10^10; dn = 10^6; db = 2*10^-16; b0 = p = 3; q = 10/3; b = q - Log[2]; f = Compile[{n, p, q}, (p*((p-5)*p + 8) + n*(n*p + (2*p-5)*p + 2) + q - 5)/((p-5)*p + n*(n + 2*p - 5) + 7)]; For[n = 3, n <= nmax, n++, If[Divisible[n, dn], b0 = b]; r = f[n, p, q]; b = r - Log[n]; p = q; q = r; If[Divisible[n, dn], Print["n = ", n, " b = ", b]; If[Abs[b - b0] < db, Break[]]]]; RealDigits[b] // First

Q(0) = 0, Q(n) = (1/2)*(1+Q(n-1)^2), Q(n) ~ 1-2/(n+log(n)+b) when n -> infinity.

Extended to 99 digits using Jon E. Schoenfield's evaluation, Sep 05 2016

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A242672 Decimal expansion of an optimal stopping constant related to the Secretary problem.

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A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

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A245771 Decimal expansion of 'b', an optimal stopping constant associated with the secretary problem when the objective is to maximize the hiree's expected quality.

Views

Author

Keywords

Examples

References

Links

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Magma

Mathematica

Formula

Extensions