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%I #16 Nov 15 2022 12:03:50
%S 1,3,8,19,45,75,135,178,209,732,1351,1907,5469,28414,37373,404055,
%T 902497
%N Exp(n) is closer to an integer than any previous exp(k) for 1 <= k < n.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 178, p. 56, Ellipses, Paris 2008.
%e a(2) = 3: exp(3) = 20.08... is closer to an integer than exp(1) = 2.718...
%e At 37373 the difference from an integer is 0.0000010493779591646530966...
%t a = 1; Do[ d = Abs[ Round[E^n] - N[E^n, Ceiling[ Log[10, E^n] + 10]]]; If[d < a, Print[n]; a = d], {n, 1, 50000}]
%o (PARI) {default(realprecision,1000); d(x)=abs(x-round(x))}; a(n)=local(m); if(n<2,n>0,n=a(n-1); m=d(exp(n)); until(d(exp(n))<m,n++); n)
%o (PARI) d(x)=x=frac(x); min(x,1-x)
%o D(n)=localbitprec(n/log(2)+99); d(exp(n))
%o r=1; for(n=1,4e4, t=D(n); if(t<r, r=t; print1(n", "))) \\ _Charles R Greathouse IV_, Oct 31 2022
%Y Cf. A000149, A001671, A004790, A080053.
%K nonn,more
%O 1,2
%A _Donald S. McDonald_, Jan 20 2003
%E Corrected and extended to 1351 by several correspondents, Jan 20 2003
%E a(12)-a(15) from _Robert G. Wilson v_, Jan 20 2003
%E a(16)-a(17) from _Charles R Greathouse IV_, Nov 01 2022
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