Surjectivity of Finite Rank-Capped Enots-Wolley Sequences

Source abstract

For each fixed integer $K\ge2$, we consider the Enots--Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots--Wolley sequence.