

A054404


Number of daughters to wait before picking in sultan's dowry problem with n daughters.


14



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
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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/(n1)).  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, Sultan's Dowry Problem.
Wikipedia, Secretary problem.


FORMULA

a(n) = the integer r that maximizes (r/n)(1/r+1/(r+1)+...+1/(n1)).  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..n1) ;
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}] (* JeanFrançois 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 * A307152 A008671 A199017
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



