list manipulation – Question about adding consecutive terms

I have a question. I can’t wrap my head around it. It is suppose to be done on Mathematica.

We have a sequence formula
Where x is positive and y is greater than once.
There is a different sequence for each value of y from 1 to 100.
For the first 10,000 terms of the sequence((for a given y), we need to find the gcd of the two consecutive terms of the sequence.

Then I’m suppose to make aa list with the max value for the GCD for each of the 100 sequences

this is what i did so far. I did GCD(a, a+1) so I can get the consecutive elements from the list, but how do I make it continue down the list? Thank you!

    f = x^2 + y^2
    l1 = f /. y -> Range(1, 100);
    l2 = GCD(l1 /. x -> Range(1, 10000));
    a = Simplify(list2)