1,2

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.

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.

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;

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}] (from Robert G. Wilson v Nov 03 2004)

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

Sequence in context: A101982 A041947 A102013 this_sequence A041299 A001112 A042079

Adjacent sequences: A079937 A079938 A079939 this_sequence A079941 A079942 A079943

more,nonn

Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 2003

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

a(16), a(17), a(18), a(19) & a(20) from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 03 2004