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Claim. Put $T(n)=\phi(n)+1$, and for a prime $p$ let $B_p$ be the set of positive integers $n$ that reach $p$ after zero or more iterations of $T$. For every odd prime $p$, $B_p=\{p,2p\}$ if and only if the only positive integer solutions of $\phi(n)=p-1$ are $n=p$ and $n=2p$. For every $\delta>0$ there is $C_\delta$ such that, for every $x\ge3$, at most $C_\delta x/(\log x)^{2-\delta}$ primes $p\le x$ have $B_p\ne\{p,2p\}$. Thus, for almost every prime $p$, exactly two positive integers reach $p$, and the set of $n$ that reach $p$ has density zero.

Covers. The second and third questions of Problem 409 at almost every prime, that is, at every prime outside an exceptional set of at most $C_\delta x/(\log x)^{2-\delta}$ primes up to $x$: only finitely many $n$ reach such a prime, and the density of the $n$ that reach it is zero. The union of these two-element basins has natural density zero among the positive integers, so almost every integer reaches an exceptional prime; the claim gives no information about the basin of any exceptional prime. Whether some prime is reached by infinitely many $n$, the density of each exceptional basin, and the first question, how many iterations are needed, remain open.

Credit. The counting input is Theorem 4.1 of Banks, Ford, Luca, Pappalardi and Shparlinski, Values of the Euler Function in Various Sequences, Monatsh. Math. 146 (2005), 1–19, which bounds the primes $p\le x$ for which $\phi(n)=p-1$ has a solution other than $p$ and $2p$; their introduction records the almost-all-primes consequence for these preimages. Plasma AI supplies the identification of the whole basin, using that $2p$ is not a value of $T$. An argument posted in the problem's forum thread on 22 July 2026 by the account ankit, recorded on its own claim page, gives the basin $\{p,2p\}$ for all but $O(\sqrt x)$ primes $p\le x$ with $p\equiv3\pmod4$, ending with the same step.

Standing. Tier 0: the deduction is author-recorded by Plasma AI, with no independent review or Lean proof. The 2005 theorem is a published input: its definition, statement and proof were read in the authors' manuscript, and the totient-counting and sieve results on which it rests were not independently audited. Catalog acceptance is pending.

Depends on. L104 (tier 0), with its external input Banks, Ford, Luca, Pappalardi and Shparlinski's Theorem 4.1.