Contined-Fraction Recursion Sequence

[Leroy Quet]

I posted the below, in part, to sci.math yesterday:

>I have written before (many times, it seems) on sci.math about
>recursively-defined continued-fractions. But now I am wondering about
>what is known about any such CFs, which are so that:
>
>If {a(k)} is a sequence of positive integers, and
>
>if [a(1); a(2), a(3),...,a(m)] = n(m)/d(m)
>
>(n(m) and d(m) are relatively prime positive integers),
>
>then each a(m) is picked so that n(m) and d(m) have some relationship
>to each other, and so that a(m) is the lowest positive integer (or
>lowest yet unpicked positive integer) that produces the properly
>related {n(m),d(m)}.
>
>An example:
>
>a(1) = 1;
>
>a(m) is the lowest positive integer such that,
>
>if [a(1); a(2), a(3),...,a(m)] = n(m)/d(m),
>
>then n(m) and d(m) have the same number of positive divisors.
>
>I may very well have made an error in calculating this sequence
>to-start by hand, but I get:
>
>1, 2, 2, 5, 1,...
>
>Does this continued fraction converge to a value with a closed-form
>expression? (Most likely, no.)

---

I asked if this sequence was already in the EIS, and Don Rebel replied, 
in part:

>I get 
>    1,2,2,5,1,1,6,1,1,15,1,1,3,2,3,2,4,7,3,1,39,1,3,38,26,6,3,2,
>    17,14,9,15,20,1,22,7,1,10,8
>
>> Does this continued fraction converge to a value with a closed-form
>> expression? (Most likely, no.)
>    I'd be amazed, unless the sequence is finite: that would make the
>    value rational. It converges to
>    1.4067389587822642355899413745452287398215142702087...
>    The Inverse Symbolic Calculator's smart lookup doesn't recognize it.
>
>> ... the EIS. Is it in the database already?
>    No.
>




Anything that those at seq.fan can add to any of this?? 

Thanks,
Leroy Quet