A080142 Greedy frac multiples of 1/Pi: a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=1/Pi, where "frac(y)" denotes the fractional part of y.

1, 2, 22, 44, 66, 88, 110, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 6390, 6745, 7100, 7455, 7810, 8165, 104348, 104703, 105058, 105413, 105768, 208696, 209051, 312689, 313044, 417037

Offset: 1

Author

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

Keywords

• nonn

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(3) = 22 since frac(1x) + frac(2x) + frac(22x) < 1, while frac(1x) + frac(2x) + frac(k*x) > 1 for all k>2 and k<22.
		

Crossrefs

Cf. A079938, A079939, A079940, A079941, etc.

Programs

Maple

Digits := 1000: a := []: s := 0: x := evalf(1.0/Pi): for n from 1 to 10000000 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, fps = Plus @@ Table[FractionalPart[a[i]*Pi^-1], {i, n - 1}]}, While[fps + FractionalPart[k*Pi^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 40}] (* Robert G. Wilson v, Nov 03 2004 *)