Dividing Numerator or Denomninator Of Harmonic Numbers

[Leroy Quet]

Two days ago I submitted this sequence to the EIS.

>A000001 1,2,2,4,4,3,6,8,9,5,3,4,12,7,5,16
>%N A000001 a(n) is lowest positive integer m such that n divides either 
>the numerator or the denominator of the (reduced) H(m) = sum{k=1 to m} 1/k.
>%e A000001 a(5) = 4 because H(4) = 25/12 is the first harmonic number with 
>either its numerator or denominator divisible by 5.
>a(6) = 3 because H(3) = 11/6 is the first harmonic number with either its 
>numerator or denominator divisible by 6.
>%Y A000001 A001008,A002805
>%O A000001 1
>%K A000001 ,more,nonn,

My question seems like it might have an obvious answer.
(Yet when I posted it to sci.math, I received no replies as of today.)
Does every positive integer divide the numerator or denominator
of at least one harmonic number?

It seems as if a(n) might always be <= n.
At least that is the way it is for all the terms above.

thanks,
Leroy Quet