Let $x$ be a given positive integer.
I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$.
Both constructive and nonconstructive results.
For clarity with longest I meant the most elements.
For example 1,3,5,7 is longer than 10,20,30.
I have no lack of Ideas relating to this but none are formal or even convincing.
For instance - inspired by sieve theory - I considered:
Longest progression $~$
$$x^2 \left(1 - \frac{1}{3} - \frac{1}{9}\right)\left(1 - \frac{1}{5} - \frac{1}{25}\right)\dots$$
(Product over odd primes up to $x$)
Or $\frac {\pi(x)}{\zeta(2)}$
Etc etc
But those are handwaving arguments.
Maybe its just a Deeper understanding of the Chinese remainder, Fermat little and quadratic reciprocity I need?