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Claim. Put $T(n)=\phi(n)+1$ for positive integers $n$; every prime is fixed by $T$. Let $F(n)$ be the number of iterations of $T$ needed to reach a prime, with $F(p)=0$ for a prime $p$, and for a prime $p$ let $B_p$ be the set of positive integers $n$ that reach $p$ after zero or more iterations of $T$. Every positive integer reaches a prime, $F(1)=1$, and

$$F(n)\le4(\sqrt n-\sqrt2)\qquad(n\ge2).$$

Moreover $B_2=\{1,2\}$, $B_3=\{3,4,6\}$, $B_5=\{5,8,10,12\}$, $B_7=\{7,9,14,15,16,18,20,24,30\}$ and $B_{11}=\{11,22\}$. Thus each of these five primes is reached by only finitely many $n$, and the set of $n$ reaching it has density zero.

Covers. The three questions of Problem 409, each in part. For the first, how many iterations are needed, an explicit bound of order $\sqrt n$, against the trivial $o(n)$ that the problem page records; it does not determine the order of growth, and no bound $o(\sqrt n)$ is established. For the second and third, the five primes $2,3,5,7,11$: none is reached by infinitely many $n$, and the $n$ reaching each have density zero. Of these only $11$ has the basin $\{p,2p\}$ of the almost-every-prime claim, so $2$, $3$, $5$ and $7$ lie in the exceptional set that claim leaves open. The lists for $5$ and $7$ answer the questions recorded in Guy's B41: the numbers leading to $5$ are exactly $5,8,10,12$, and those leading to $7$ exactly $7,9,14,15,16,18,20,24,30$. A prime reached by infinitely many $n$, if one exists, is at least $13$. Whether one exists, the density of every other basin and the order of growth of the number of iterations remain open, and the problem remains open.

Credit. Wouter van Doorn (account Woett) observed in a comment on the problem's forum thread on 22 July 2026 (its claim page) that $\phi(n)\le n-\sqrt n$ for composite $n$ gives $F(n)=O(\sqrt n)$. The notes the account ankit linked the same day (its claim page) bound the number of iterations by $5\sqrt n+1$ and report the basin of every prime below $30000$ computed exactly and finite, with the same five basins listed here; neither was checked. The descent here uses the same inequality with the constant $4$. The questions about $5$ and $7$ are Finucane's, recorded in B41 of Guy's Unsolved Problems in Number Theory. The descent and the inverse-tree calculation are Plasma AI's. Neither result was compared with literature beyond the problem page's search.

Standing. Tier 0: both arguments are author-recorded by Plasma AI, without independent review or Lean proof. They are written arguments with no retained evidence program; the basin page gives the finite enumeration that proves each inverse-totient list complete. Catalog acceptance is pending.

Depends on. the small-basins lead (tier 0), with its descent and inverse-tree calculation.