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Minimal exponents m such that the fractional part of e^m obtains a minimum (when starting with m=1).
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%I #9 Mar 23 2019 17:58:22

%S 1,2,3,9,29,45,75,135,219,732,1351,3315,4795,4920,5469,28414,37373

%N Minimal exponents m such that the fractional part of e^m obtains a minimum (when starting with m=1).

%C Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of e^m is less than the fractional part of e^k for all k, 1<=k<m.

%C The next such number must be greater than 100000.

%C a(18) > 300,000. _Robert Price_, Mar 23 2019

%F Recursion: a(1):=1, a(k):=min{ m>1 | fract(e^m) < fract(e^a(k-1))}, where fract(x) = x-floor(x).

%e a(4)=9, since fract(e^9)=0.08392..., but fract(e^k)>=0.08553... for 1<=k<=8; thus fract(e^9)<fract(e^k) for 1<=k<9.

%t $MaxExtraPrecision = 100000;

%t p = 1; Select[Range[1, 300000],

%t If[FractionalPart[E^#] < p, p = FractionalPart[E^#]; True] &] (* _Robert Price_, Mar 23 2019 *)

%Y Cf. A153661, A153669, A153677, A153685, A153693, A153705, A154130, A137994, A153717, A000149.

%K nonn,more

%O 1,2

%A _Hieronymus Fischer_, Jan 06 2009