COMMENT on A114717. Increasing compositeness?

[Mitch Harris]

Antti Karttunen wrote:
>
> Now it would be also interesting to compute the positions (and values) 
> where this sequence A114717
> obtains A) distinct new values (i.e. 1,2,5,14,48,42,2452,462, etc. and 
> the positions where they occur, i.e. 1,6,12,24,30,36,...)
> and B) records (i.e. 1,2,5,14,48,2452, and their positions: 
> 1,6,12,24,30,60, ....)
> and does any of these four sequences occur already in OEIS, 

A quick search gets nothing. The "value" sequences will be just as hard 
(if not harder) to compute as the original. As to the "records" sequences...

> and if not, 
> then whether they give
> us a yet another useful measure of "increasing compositeness" ?

I would think that the factorization would be the best measure of 
compositeness (or even just the exponents in the factorization, which is 
all that the # of linear extensions of the divisor lattice depends on). 
Like the example in the comments, a(12) = a(75) because both have the 
factorization exponent signature of (2, 1), so I would say that 12 and 
75 are equally 'composite'.

The set of numbers that are the least in their equivalence class of 
exponent signatures is a related idea. So 12 would be in the set, but 
not 18, 20, 45, 75, etc.

See A025487: Least integer of each prime signature

Mitch