AI 中文总结
本文证明对任意固定整数K≥2，满足恰有2到K个不同素因子的Enots-Wolley序列是满射的，即所有满足该限制的整数都会出现，通过分析素支撑的有限选择与Landau估计的矛盾完成证明。
AI 中文摘要
对于每个固定整数$K\ge2$，我们考虑Enots--Wolley序列，其中在初始项$1,2$之后的每一项被要求具有2到$K$个不同的素因子。我们证明满足此限制的每个整数都会出现。如果某个精确的素支撑$T$仅被有限次选择，那么在有限截断之后，满足$T$的项将形成短片段，并且每个完整项将迫使在可比较的数值高度上出现一个更早的恰当项。$|T|=K$的情况随后被直接排除。在剩余情况$|T|<K$中，在记录恰当值$H$处，贪心性迫使每个低于$H$的未阻塞秩-$K$整数（恰好包含$T$的一个素数）已经更早出现。固定阶Landau估计表明，此恰当群体的阶为$H(\log\log H)^{K-2}/\log H$，而同一尺度上整个可能的完整群体具有严格更小的对数阶。这一矛盾证明了满射性。该定理涉及每个固定秩封顶，并未解决无限制Enots--Wolley序列的满射性。
英文摘要
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.
