a step beyond Euclid and Fermat, 4

I've introduced the Euclid sequence e(n), and Fermat introduced his Fermat sequence f(n). Now let me introduce sequence g(n), which again has pairwise relatively prime terms. In another post I will show that sequence g is essentially different from sequences e and f and from all sequences euf. It is also satisfying that the terms g(n) are asymptotically smaller than e(n) and f(n), that they are of the order of the square root of e(n) or of f(n).

First, let me introduce an auxiliary sequence A(n):

• A(1) := 3
• A(2) := 10
• A(n+1) := A(n-1) ⋅ (A(n) - A(n-1))     ∀ n= 2 3 ...

so that A(3) = 3*(10-3) = 21, A(4) = 10*(21-10) = 110, etc.

It is not too hard to see that

A(n) = n mod 2     ∀ n=1 2 ...

Now let

• g(0) := 2
• g(1) := 3
• g(2) := 5
• g(n+1) := A(n) - A(n-1)     ∀ n=2 3 ...

so that g(3) = 7, g(4) = 11, g(5) = 89, etc. Also:

• A(n+1) := A(n-1) ⋅ g(n+1)     ∀ n= 2 3 ...

It is a fascinating topic to me but I am tired now. I'll continue later.