A137315 related to A139795 ?

[Richard Mathar]

I am wondering:

Consider the continued fractions of the harmonic
numbers,
H(n) = sum{k=1 to n} 1/k.

(A100398 gives the terms of the continued
fractions. A055573 gives the number of terms in
each continued fraction.)

I wanted to submit the sequence of positive
integers n such that the simple continued
fraction of each H(n) contains all distinct
terms.

Using only the terms given in A100398, I get that
the sequence begins: 1,2,4,... (There are no
other n's <= 15.)
My instinct is that there are only a finite
number of terms in this sequence.
But I also have a small doubt of the sequence's
finitude.

Could someone please calculate more terms?

Thanks,
Leroy Quet