## 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).

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