A208530 - OEIS

%I #16 Oct 26 2024 03:32:21

%S 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,

%T 678,685,1356,1363,2034,2041,2719,3397,4075,4753,5431,18412,19090,

%U 19768,22487,23165,23843,41577,42255,60667,61345,102922,103600,122012,122690

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

%C 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.

%H Robert G. Wilson v, <a href="/A208530/b208530.txt">Table of n, a(n) for n = 1..66</a>

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

%p nm:= x -> abs(x-round(x)):

%p f:= n -> is(max(nm(n*Pi),nm(n*exp(1)))<n^(-1/2)):

%p select(f, [$1 .. 20000]);

%t 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 *)

%K nonn

%O 1,2

%A _Robert Israel_, Feb 27 2012