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

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

Offset: 1

Author

José María Grau Ribas, Feb 18 2019

Keywords

• nonn

Comments

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

Examples

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

Links

• 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

Crossrefs

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

Programs

Mathematica

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]

Formula

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

Extensions

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