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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.

%I A282025 #32 Feb 24 2017 02:52:06
%S A282025 3,8,13,18,23,27,32,37,42,47,52,57,62,67,72,77,82,86,91,96,101,106,
%T A282025 111,116,121,126,131,136,141,146,150,155,160,165,170,175,180,185,190,
%U A282025 195,200,205,210,214,219,224,229,234,239,244,249,254,259,264,269,273,278,283
%N 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.
%C A282025 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<n. These best values are r=1 for n<=8, r=2 for n<=13, r=3 for n<=18 and so on.
%C A282025 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.
%C A282025 We added numbers 27, 86 and 91 that are apparently missing in the preprint. _R. J. Mathar_, Feb 22 2017
%H A282025 Vincenzo Librandi, <a href="/A282025/b282025.txt">Table of n, a(n) for n = 0..202</a>
%H A282025 L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, <a href="https://arxiv.org/abs/1603.03928">A variant of the Secretary Problem: the Best or the Worst</a>, arXiv preprint arXiv:1603.03928 [math.PR], 2016.
%p A282025 P := proc(n,
%p A282025     option remember;
%p A282025     local i;
%p A282025     2*r/n*((r/n-1)+add(1/i,i=r..n-1)) ;
%p A282025 end proc:
%p A282025 Pmax := proc(n)
%p A282025     option remember;
%p A282025     local r;
%p A282025     for r from 1 to n-1 do
%p A282025         if P(n,r+1) < P(n,r) then
%p A282025             return r ;
%p A282025         end if;
%p A282025     end do:
%p A282025 end proc:
%p A282025 A282025 := proc(r)
%p A282025     local n ;
%p A282025     if r = 0 then
%p A282025         return 3;
%p A282025     end if;
%p A282025     for n from r+1 do
%p A282025         if Pmax(n+1) = r+1 then
%p A282025             return n;
%p A282025         end if;
%p A282025     end do:
%p A282025     return -1 ;
%p A282025 end proc:
%p A282025 seq(A282025(r),r=0..80) ; # _R. J. Mathar_, Feb 22 2017
%t A282025 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 *)
%Y A282025 Cf. A106533, A054404.
%K A282025 nonn
%O A282025 0,1
%A A282025 _N. J. A. Sloane_, Feb 11 2017