Zeisel Numbers

Helmut Zeisel found that p = 2^(k-1) + k is prime if k=1885.  The
number 1885 is also interesting for another reason:  It can be 
factored as  1885 = (1)(5)(13)(29).  Notice that

                       (1)
                       (5) =  2(1) + 3
                      (13) =  2(5) + 3
                      (29) = 2(13) + 3

This could be generalized to any number N = (p_1)(p_2)...(p_k)
such that  
                    p_n  =  A * p_(n-1)  +  B

where A and B are constants and we define  p_0 = 1.  The number 114985 
is also a "Zeisel Number", because 2(29)+3=61.  However, that's the end 
of the line for {A=2,B=3}, because 2(61)+3=125.

More generally, we could define higher order Zeisel Numbers as integers
whose prime factors satisfy any dth order linear recurrence, with the
"initial values" p_0,..,p_(d-1) = 1.  As an example of a second order 
Zeisel Number, consider an integer N having prime factors p_i such that

                   p_n  =  2 * p_(n-1)  +  p_(n-2)   

with p_0 = p_1 = 1.   This would give the number

                   14637  =  [1] [1] (3) (7) (17) (41)

If, in addition to requiring the initial values of 1, we consider only 
the "completed" Zeisel Numbers (i.e., those for which the next p_i 
produced by the recurrence is composite), then there is a unique Zeisel 
Number for any given linear recurrence.  For example, the Zeisel Number 
for the Fibonacci recurrence is 6 = [1][1](2)(3).  Hey, a perfect 
number!  But I digress...