A079940 Greedy fractional multiples of 1/e: a(1)=1, Sum_{n>0} frac(a(n)/e) = 1.

1, 3, 4, 11, 87, 193, 386, 579, 1457, 23225, 49171, 98342, 147513, 196684, 566827, 13580623, 28245729, 56491458, 84737187, 112982916, 438351041, 466596770, 494842499

Offset: 1

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

• more
• nonn
• changed

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.

Examples

			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.
		

Links

• K. Girstmair, On the Asymptotic Behavior of Dedekind Sums, J. Int. Seq. 17 (2014) # 14.7.7, example 2.

Crossrefs

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

Programs

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 *)

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

a(21)-a(23) from Sean A. Irvine, Aug 30 2025