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

1,2

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.

Links

Table of n, a(n) for n=1..40.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A221762 A154798 A350318 * A364484 A306969 A200946

Adjacent sequences: A080139 A080140 A080141 * A080143 A080144 A080145

Keyword

nonn

Author

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

Status

approved