A054404 -id:A054404 - cp's OEIS frontend

A225676 Numbers n such that A054404(n) does not equal round(n/e).

7, 10, 15, 18, 26, 29, 34, 37, 45, 48, 56, 64, 67, 75, 83, 86, 94, 97, 102, 105, 113, 116, 121, 124, 132, 135, 140, 143, 151, 154, 162, 170, 173, 181, 189, 192, 200, 203, 208, 211, 219, 222, 227, 230, 238, 241, 249

Offset: 1

José María Grau Ribas, May 12 2013

nonn

Cf. A054404, A225593.

PR[n_, r_] := (r/n)*Sum[1/k, {k, r, n - 1}]; maxi[li_] := {Do[If[li[[n + 1]] < li[[n]], aux = n; Break[]], {n, 1, Length[li] - 1}], aux}[[2]]; SEQ[1] = 0; SEQ[2] = 1; SEQ[n_] := maxi[Table[PR[n, i], {i, 1, n - 1}]]; Select[Range@400, ! SEQ[#] == Round[#/E] &]

A306480 Numbers k such that A054404(k) is not floor(k/e - 1/(2*e) + 1/2).

Original entry on oeis.org

97, 24586, 14122865, 14437880866, 23075113325617, 53123288947296842, 166496860519928411041, 681661051602157413173890, 3532450008306093939076231361, 22600996284275635202947629995722, 174979114331029936735527491233938577, 1612273088535187752419835130130200398626

Offset: 1

José María Grau Ribas, Feb 18 2019

nonn

Numbers k such that the optimal threshold in the secretary problem with k candidates is not floor(k/e - 1/(2*e) + 1/2).

			A054404(97)=35 but floor(97/e - 1/(2e) + 1/2) = 36.

J. P. Gilbert and F. Mosteller, Recognizing the Maximum of a Sequence, Journal of the American Statistical Association, Vol. 61 No. 313 (1966), 35-73.
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A001113, A054404, A226242, A226243, A226082.

P[r_, n_] := If[r == 0, 1/n, r/n (PolyGamma[0, n] - PolyGamma[0, r])]
in[n_] := (n - 1/2)/E + 1/2 - (3E - 1)/2/(2 n + 3E - 1) - 1
su[n_] := n/E - 1/2/E + 1/2
A054404[n_] := If[P[Floor[su[n]], n] >= P[Ceiling[in[n]], n], Floor[su[n]], Ceiling[in[n]]]
lista = Select[Range[25000], ! Floor[su[#]] == Ceiling[in[#]] &];
IS[n_] := If[Floor[su[n]] == Ceiling[in[n]], False, ! (A054404[n] == Floor[su[n]])]
Select[lista, IS]

Empirical observation: a(n) = (2*d(6k+3)+1)/2, where d(m) is the denominator of the truncated continued fraction [a_0;a_1,a_2,...,a_m] of 1/e. - Giovanni Corbelli, Jul 23 2021

a(4)-a(12) from Jon E. Schoenfield, Feb 28 2019

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

Original entry on oeis.org

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...).

A226242 Numerators of the probability of success in sultan's dowry problem with n daughters.

Original entry on oeis.org

1, 1, 1, 11, 13, 77, 29, 459, 341, 3349, 251, 32891, 28271, 395243, 420983, 74587, 158183, 2833255, 853661, 3407275, 77976391, 27223837, 28399557, 1814074083, 5665315119, 29397421371, 27452509171, 85332099113, 88200436013

Offset: 1

José María Grau Ribas, Jun 01 2013

			1, 1/2, 1/2, 11/24, 13/30, 77/180, 29/70, 459/1120, ...

Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.
Wikipedia, Secretary problem.

Cf. A226243 (denominators), A054404.

G[k_, n_] := G[k, n] = 1/( k + 1) Max[(k + 1)/n, G[k + 1, n]] + k/(k + 1)G[k + 1, n]; G[n_, n_] = 0; Numerator@Table[G[0, n], {n, 1, 20}]

a(n)={my(g=0); forstep(k=n-1, 0, -1, g = max(1/n, g/(k+1)) + k*g/(k+1)); numerator(g)} \\ Andrew Howroyd, Nov 12 2018

A226243 Denominators of the probability of success in sultan's dowry problem with n daughters.

Original entry on oeis.org

1, 2, 2, 24, 30, 180, 70, 1120, 840, 8400, 630, 83160, 72072, 1009008, 1081080, 192192, 408408, 7351344, 2217072, 8868288, 203693490, 71131060, 74364290, 4759314560, 14872858000, 77338861600, 72282089880

Offset: 1

José María Grau Ribas, Jun 01 2013

			1, 1/2, 1/2, 11/24, 13/30, 77/180, 29/70, 459/1120, ...

Andrew Howroyd, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.
Wikipedia, Secretary problem.

Cf. A226242(numerators), A054404.

G[k_, n_] := G[k, n] = 1/( k + 1) Max[(k + 1)/n, G[k + 1, n]] + k/(k + 1)G[k + 1, n]; G[n_, n_] = 0; Denominator@Table[G[0, n], {n, 1, 20}]

a(n)={my(g=0); forstep(k=n-1, 0, -1, g = max(1/n, g/(k+1)) + k*g/(k+1)); denominator(g)} \\ Andrew Howroyd, Nov 12 2018

A242673 Decimal expansion of the unique real solution of the equation Ei(x)-gamma-log(x) = 1, where Ei is the exponential integral function and gamma the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

This constant is an auxiliary constant used in computing the asymptotic probability of success in one of the Secretary problems.

			0.804352262845637584654638587784070551...

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, A242674.

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

A226033 Round(n * exp(-1 - 1/(2n))), an approximation to the number of daughters to wait before picking in the sultan's dowry problem (Better that A225593).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

José María Grau Ribas, May 24 2013

nonn

It is very similar to the sequence A054404, but differs for example at n=97 (see A226082).

Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.

Cf. A054404, A225593.

A226033 := proc(n)
    round(n/exp(1+1/2/n)) ;
end proc: # R. J. Mathar, Jun 09 2013

Mathematica
```
Table[Round[n*E^(-1-1/(2*n))], {n,100}]
```

A226082 Number of daughters to wait before picking in the sultan's dowry problem that is not equal to round(ne^(-1 - 1/(2n))).

Original entry on oeis.org

97, 1361, 24586, 73757

Offset: 1

José María Grau Ribas, May 25 2013

Numbers n such that A054404(n) is not equal to A226033(n).

Cf. A054404, A226033.

li[n_] := -1/2* 1/ProductLog[-Exp[1 - 1/((2  n - 2))]/(2n-2)]; Table[If[! Round[n*E^(-1 - 1/(2*n))] == Ceiling[li[n]], Print[n]; n], {n, 4, 1000000}]

A243533 Decimal expansion of 'c', an asymptotic constant related to a variation of the "Secretary problem" with a uniform distribution.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2014

			1.3531302722959332816529440324922596269...

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

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

Cf. A054404, A242672.

digits = 65; c = 2*NSum[Log[k]/(k^2 - 1), {k, 3, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 10^4, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 10, Method -> "DoubleExponential"}}] - (Log[2]/3); RealDigits[c, 10, digits] // First

log(A242672).

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

cp's OEIS Frontend

A225676 Numbers n such that A054404(n) does not equal round(n/e).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A306480 Numbers k such that A054404(k) is not floor(k/e - 1/(2*e) + 1/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A226242 Numerators of the probability of success in sultan's dowry problem with n daughters.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A226243 Denominators of the probability of success in sultan's dowry problem with n daughters.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A242673 Decimal expansion of the unique real solution of the equation Ei(x)-gamma-log(x) = 1, where Ei is the exponential integral function and gamma the Euler-Mascheroni constant.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A226033 Round(n * exp(-1 - 1/(2n))), an approximation to the number of daughters to wait before picking in the sultan's dowry problem (Better that A225593).

Views

Author

Keywords

Comments

A226082 Number of daughters to wait before picking in the sultan's dowry problem that is not equal to round(ne^(-1 - 1/(2n))).