1,5,19,13,211,7,2059,97,1009 - of interest?

[Gottfried Helms]

Hi,

 fiddling with the factorizing of mersenne numbers (2^n-1) and
 the 3^n-2^n-differences I focused the sequences of *new* factors,
 which occur on each ascending step from n to n+1.
 For 3^n-2^n it is the sequence  (starting at n=0)

     A(n)=   0,1,5,19,13,211,7,2059,97,1009,...

 where the full factorization is multiplicative; meaning that
 the composition of factors is determined by the prime-facto-
 rization of n.

 Let n be 7 then the factors of

  f(n,a,b) := (a^n-b^n)/(a-b)

 with the shorthand for the 3^n-2^n-example

  g(n) := f(n,3,2)

 is then, for instance,

  g(7)  = A(7) =  2059   // since n is prime

 let n be 3 then the factors of

  g(3) = A(3) = 19      // since n is prime

 let n be 21, then the factors are

  g(21) = A(3)*A(7)*A(21)

 and whether n is composite or not, with each n (at least)
 one new factor occurs besides the factors determined by
 the primefactors of n - so it is not *purely* multiplicative.

 This seems connected to the understanding of the Lucas-
 Lehmer-test, if I got things right.
 So this sequence may be of interest - on the other hand
 I doubt it is of interest to add sequences for each
 combination of a and b in f(n,a,b) .. ;-)

Gottfried Helms