On powers of the diophantine function $\star :x\mapsto x(x+1)$

We treat the functions ${\star }^k:\mathbf{N}\to \mathbf{N}$ where $\star :x\mapsto \star x:=x(x+1)$. The set $\{{\star }^{k}x+1:\{x,k\}\subseteq \mathbf{N}\}$ is pairwise coprime; so, the set $\mathbf{P}$ of primes is infinite. Our Theorem 4 resorts to the mother sequence, M, that is obtained by factoring the infinite sequence $2,3,4,5,\dots$ into prime powers. For each $x\ge 1$ we define the gross $x$-sequence, ${\gamma }_{\star }(x):=⟨x+1;\star x+1;{\star }^{2}x+1;{\star }^{3}x+1;\dots ⟩$, and also the star sequence, $x^{\star }$, obtained by factoring the terms of ${\gamma }_{\star }(x)$ into prime powers. It turns out that ${\gamma }_{\star }(1)$ is Sylvester’s sequence, A00058 in the Online Encyclopedia of Integer Sequences, OEIS, and that ${\gamma }_{\star }(2)$ is the sequence A082732 in the OEIS. Theorem 3. For every integer $x\ge 1$ there is a prime $p(x)$ that divides no member of $\{{\star }^{k}x+1:k\ge 0\}$. Theorem 4. For each sequence $\eta$ of powers of primes there are infinitely many subsequences $c$ of M such that numerically $\eta =c_j$ but where the term-set family in M of those $c_j$ is formally. pairwise disjoint. Theorem 6. $1\mathrm{/}x={\sum }_{k=0}^{n-1}1\mathrm{/}({\star }^{k}x+1)+1\mathrm{/}({\star }^{n}x)={\sum }_{k=0}^{\mathrm{\infty }}1\mathrm{/}({\star }^{k}x+1)$ for all $\{x,n\}\subseteq \mathbf{N}$. Theorem 7. For every $x\in \mathbf{N}$, when $x^{\star }:=⟨x_j{⟩}_{j=0}^{\mathrm{\infty }}$ then ${\sum }_{j=0}^{\mathrm{\infty }}1\mathrm{/}{x}_j=\mathrm{\infty }$.