A079939 Greedy frac multiples of e: a(1)=1, Sum_{n>0} frac(a(n)*e)=1.

1, 3, 7, 14, 39, 78, 394, 1001, 2002, 3003, 9545, 10546, 27634, 154257, 398959, 797918, 1196877, 1595836, 5394991, 5793950, 15786014, 130087267, 312129649, 624259298

Offset: 1

Author

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

Keywords

• more
• nonn
• changed

Comments

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

Examples

			a(4) = 14 since frac(1x) + frac(3x) + frac(7x) + frac(14x) < 1, while frac(1x) + frac(3x) + frac(7x) + frac(k*x) > 1 for all k>7 and k<14.
		

Crossrefs

Cf. A007677 (denominators of convergents to e), A079934, A079937, A079940.

Programs

Maple

Digits := 100: a := []: s := 0: x := 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;

Extensions

a(15)-a(16) from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003

a(17)-a(24) from Sean A. Irvine, Aug 30 2025