A259483 - OEIS

%I #10 Jul 15 2015 03:41:50

%S 1,2,3,4,7,9,10,19,20,29,39,42,52,62,72,91,101,111,163,254,264,365,

%T 629,730,994,1095,1359,1460,1724,2089,2454,2819,9076,9441,9806,18517,

%U 18882,39488,48929,58370,67811,107299,116740,156228,165669,214598,224039,272968,331338,380267

%N Numbers n such that both n*log(2) and n*log(3) 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 Israel and Robert G. Wilson v, <a href="/A259483/b259483.txt">Table of n, a(n) for n = 1..78</a>

%e |52*log(2) - 41| and |52*log(3) - 65| are both less than 1/sqrt(52) so 52 is in the sequence.

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

%t fQ[n_] := Abs[n*Log[2] - Round[n*Log[2]]] Sqrt[n] < 1 && Abs[n*Log[3] - Round[n*Log[3]]] Sqrt[n] < 1; Select[ Range@ 400000, fQ@ # &]

%Y Cf. A208530.

%K nonn

%O 1,2

%A _Robert Israel_ and _Robert G. Wilson v_, Jun 28 2015