A079940 - OEIS

%I #16 Jan 01 2024 13:27:46

%S 1,3,4,11,87,193,386,579,1457,23225,49171,98342,147513,196684,566827,

%T 13580623,28245729,56491458,84737187,112982916

%N Greedy fractional multiples of 1/e: a(1)=1, sum_{n>0} fractional_part(a(n)/e) = 1.

%C The n-th greedy fractional multiple of x is the smallest integer m that does not cause sum(k=1..n,frac(m*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of 1/e.

%C After a(20), there is only 109305220 - 297122396/e = ~1.06317354345346734...*10^-8 to work with.

%H K. Girstmair, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Girstmair/girstmair12.html">On the Asymptotic Behavior of Dedekind Sums</a>, J. Int. Seq. 17 (2014) # 14.7.7, example 2.

%e a(4) = 11 since frac(1x) + frac(3x) + frac(4x) + frac(11x) < 1, while frac(1x) + frac(3x) + frac(4x) + frac(k*x) > 1 for all k>4 and k<11.

%p Digits := 100: a := []: s := 0: x := 1.0/exp(1.0): for n from 1 to 1000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

%t a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, ps = Plus @@ Table[ FractionalPart[ a[i]*E^-1], {i, n - 1}]}, While[ ps + FractionalPart[k*E^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 20}] (* _Robert G. Wilson v_, Nov 03 2004 *)

%Y Cf. A007676 (numerators of convergents to e), A079934, A079939, A079941.

%K more,nonn

%O 1,2

%A _Benoit Cloitre_ and _Paul D. Hanna_, Jan 21 2003

%E More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003

%E a(16)-a(20) from _Robert G. Wilson v_, Nov 03 2004