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A054404 Number of daughters to wait before picking in sultan's dowry problem. 11
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, 27, 27, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The correct rule can be found in the Gardner reference (p. 60) and in the Wikipedia article (see link): if the number of candidates is n, then the optimal r (the number of candidates to skip) is the r that maximizes (r/n)(1/r+1/(r+1)+...+1/(n-1)). - Zvi Mendlowitz (zvi113(AT)zahav.net.il), Jul 12 2007

REFERENCES

M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1000

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

Wikipedia, Secretary problem.

FORMULA

a(n) = the integer r that maximizes (r/n)(1/r+1/(r+1)+...+1/(n-1)) - Zvi Mendlowitz (zvi113(AT)zahav.net.il), Jul 12 2007

MAPLE

A054404 := proc(n)

    local r ;

    r := 0 ;

    sr := 0 ;

    for s from 1 to n do

        p := s/n*add(1/i, i=s..n-1) ;

        if p > sr then

            r := s ;

            sr := p ;

        end if;

    end do;

    return r;

end proc: # R. J. Mathar, Jun 09 2013

MATHEMATICA

a[n_] := r /. Last[ Maximize[ {(r/n)*Sum[1/k, {k, r, n - 1}], 0 <= r < n/2}, r, Integers]]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Dec 13 2011, after Zvi Mendlowitz *)

(* The code above may not work in Mma 8 *)

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}]];

Table[SEQ[n], {n, 1, 133}]  (* José María Grau Ribas, May 11 2013 *)

a[1]=0; a[2]=1; a[n_] := Block[{r}, r /. Last@ Maximize[{(r/n) * (PolyGamma[0, n] - PolyGamma[0, r]), 1 <= r < n/2}, r, Integers]]; Array[a, 75] (* Giovanni Resta, May 11 2013 *)

CROSSREFS

Sequence in context: A077219 A026405 A226033 * A008671 A199017 A189709

Adjacent sequences:  A054401 A054402 A054403 * A054405 A054406 A054407

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

EXTENSIONS

Corrected by Zvi Mendlowitz (zvi113(AT)zahav.net.il), Jul 12 2007

STATUS

approved

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Last modified July 31 13:50 EDT 2014. Contains 245085 sequences.