OFFSET
1,2
COMMENTS
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.
After a(20), there is only 109305220 - 297122396/e = ~1.06317354345346734...*10^-8 to work with.
LINKS
K. Girstmair, On the Asymptotic Behavior of Dedekind Sums, J. Int. Seq. 17 (2014) # 14.7.7, example 2.
EXAMPLE
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.
MAPLE
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;
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Benoit Cloitre and Paul D. Hanna, Jan 21 2003
EXTENSIONS
More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003
a(16)-a(20) from Robert G. Wilson v, Nov 03 2004
STATUS
approved