On a class of relatively prime sequences

For each natural number n, let a₀(n) = n, and if a₀(n),…,a_i(n) have already been defined, let a_i+1(n) > a_i(n) be minimal with (a_i+1(n), a₀(n) … a_i(n)) = 1. Let g(n) be the largest a_i(n) not a prime or the square of a prime. We show that g(n) ～ n and that $\text{g(n) > n + cn}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{log}\text{(n)}$ for some c > 0. The true order of magnitude of g(n) ? n seems to be connected with the fine distribution of prime numbers. We also show that “most” a_i(n) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our proofs: For every sufficiently large n there is a prime $\text{p < n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ with $\text{n}\text{p}$] composite.