The Wikipedia's article for Prime number shows a known and curious formula for primes from its section Formula for primes, I say the Mills' theorem (please see also the Wikipedia Mills' constant).
Question. I wondered if one can to determine or calculate a constant and choose an arithmetic function, and using a floor or ceiling function, write a formula producing square-free integers in the same way that Mill's formula is a prime-generating formula. Many thanks.
Thus that I evoke is try to write a formula and try to determine unconditionally the constant. This formula should to generate a square-free integer for each $n\geq n_0$, for certain positive integer $n_0$. I would like to know if it is possible/feasible for a similar nice arithmetic function, see the exponential of Mill's formula (I don't know what is the statement of Wright's theorem, thus I am asking about a formula similar than Mill's formula, now for integers without repeated prime factors).
I don't know if my Question is in the literature. If in the literature there is such formula explicitly comment it refering the literature, and I try to search and reat it from the literature.