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%I #43 Aug 18 2021 16:32:27
%S 97,24586,14122865,14437880866,23075113325617,53123288947296842,
%T 166496860519928411041,681661051602157413173890,
%U 3532450008306093939076231361,22600996284275635202947629995722,174979114331029936735527491233938577,1612273088535187752419835130130200398626
%N Numbers k such that A054404(k) is not floor(k/e - 1/(2*e) + 1/2).
%C Numbers k such that the optimal threshold in the secretary problem with k candidates is not floor(k/e - 1/(2*e) + 1/2).
%H J. P. Gilbert and F. Mosteller, <a href="https://www.jstor.org/stable/2283044">Recognizing the Maximum of a Sequence</a>, Journal of the American Statistical Association, Vol. 61 No. 313 (1966), 35-73.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Secretary_problem">Secretary problem</a>
%F 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
%e A054404(97)=35 but floor(97/e - 1/(2e) + 1/2) = 36.
%t P[r_, n_] := If[r == 0, 1/n, r/n (PolyGamma[0, n] - PolyGamma[0, r])]
%t in[n_] := (n - 1/2)/E + 1/2 - (3E - 1)/2/(2 n + 3E - 1) - 1
%t su[n_] := n/E - 1/2/E + 1/2
%t A054404[n_] := If[P[Floor[su[n]], n] >= P[Ceiling[in[n]], n], Floor[su[n]], Ceiling[in[n]]]
%t lista = Select[Range[25000], ! Floor[su[#]] == Ceiling[in[#]] &];
%t IS[n_] := If[Floor[su[n]] == Ceiling[in[n]], False, ! (A054404[n] == Floor[su[n]])]
%t Select[lista, IS]
%Y Cf. A001113, A054404, A226242, A226243, A226082.
%K nonn
%O 1,1
%A _José María Grau Ribas_, Feb 18 2019
%E a(4)-a(12) from _Jon E. Schoenfield_, Feb 28 2019
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