A282025 a(r) is the maximum number of secretaries for which the first r should be rejected, if selecting the one with the highest or lowest ranking are both considered a success.

3, 8, 13, 18, 23, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259, 264, 269, 273, 278, 283

Offset: 0

Author

N. J. A. Sloane, Feb 11 2017

Keywords

• nonn

Comments

According to Bayon et al, the probability P(n,r) = 2*r*((r/n-1)+sum_{i=r..n-1} 1/i)/n of success in a generalized Secretary problem for a given number n of applicants has a maximum at some value of r, 1<=r

The Beatty sequence of A106533, b(n) = floor(n*A106533), is a good approximation to r for large n. So the indices n-1 of the steps where b(n) = b(n+1)-1 are an approximation to this sequence.

We added numbers 27, 86 and 91 that are apparently missing in the preprint. R. J. Mathar, Feb 22 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..202
• L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, A variant of the Secretary Problem: the Best or the Worst, arXiv preprint arXiv:1603.03928 [math.PR], 2016.

Crossrefs

Cf. A106533, A054404.

Programs

Maple

P := proc(n,
    option remember;
    local i;
    2*r/n*((r/n-1)+add(1/i,i=r..n-1)) ;
end proc:
Pmax := proc(n)
    option remember;
    local r;
    for r from 1 to n-1 do
        if P(n,r+1) < P(n,r) then
            return r ;
        end if;
    end do:
end proc:
A282025 := proc(r)
    local n ;
    if r = 0 then
        return 3;
    end if;
    for n from r+1 do
        if Pmax(n+1) = r+1 then
            return n;
        end if;
    end do:
    return -1 ;
end proc:
seq(A282025(r),r=0..80) ; # R. J. Mathar, Feb 22 2017

Mathematica

P[n_, r_] := 2 r ((r/n - 1) + Sum[ 1/i, {i, r, n - 1}])/n; Function[s, {3}~Join~Map[-1 + Position[s, #][[1, 1]] &, Range@ Max@ s]]@ Map[Length@ TakeWhile[#, # == 0 &] &, Table[If[P[n, k + 1] < P[n, k], k, 0], {n, 300}, {k, n - 1}]] (* Michael De Vlieger, Feb 22 2017, after Maple *)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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