A208530 Numbers n such that both n*Pi and n*e are within 1/sqrt(n) of integers.

1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 21, 22, 28, 29, 35, 36, 43, 50, 57, 64, 71, 78, 85, 92, 671, 678, 685, 1356, 1363, 2034, 2041, 2719, 3397, 4075, 4753, 5431, 18412, 19090, 19768, 22487, 23165, 23843, 41577, 42255, 60667, 61345, 102922, 103600, 122012, 122690

Offset

1,2

Comments

For any irrational x and y there exist infinitely many positive integers n such that max(|n*x - Z|,|n*y - Z|) < 1/sqrt(n), where Z is the set of integers.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..66

Example

|50*Pi - 157| and |50*e - 136| are both less than 1/sqrt(50) so 50 is in the sequence.

Maple

nm:= x -> abs(x-round(x)):
f:= n -> is(max(nm(n*Pi), nm(n*exp(1)))<n^(-1/2)):
select(f, [$1 .. 20000]);

Mathematica

fQ[n_] := Abs[n*Pi - Round[n*Pi]] < 1/Sqrt[n] && Abs[n*E - Round[n*E]] < 1/Sqrt[n]; Select[Range@ 20000, fQ@# &] (* Robert G. Wilson v, Mar 10 2013 *)

Crossrefs

Sequence in context: A032870 A023763 A032904 * A088416 A057913 A245802

Adjacent sequences: A208527 A208528 A208529 * A208531 A208532 A208533

Keyword

nonn

Author

Robert Israel, Feb 27 2012

Status

approved