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Updated Oct 7

Problem 414

Statement. Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$ ) and $h_k(n)=h(h_{k-1}(n))$ . Is it true, for any $m,n$ , there exist $i$ and $j$ such that $h_i(m)=h_j(n)$ ?

Status. Open.

Source. erdosproblems.com/414, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #414, https://www.erdosproblems.com/414.

Formalization. Statement in formal-conjectures.

Progress

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Linked library material

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Linked from (19)

Arithmetic Functions arithmetic_functions/li_2026_square_annular_dynamics_coalescence_frontiers_n Corollary 5.10 (p. 10): liminf |A_k(E_k)| = 1 would imply coalescence for n + tau(n)Proposition 5.1 (p. 8): the image of the transfer map A_k is the exit set E_(k+1)Proposition 5.4 (p. 9): the crossing set over k^2 is k^2 + E_k, so R(X) <= |E_k| for X < k^2 Proposition 5.8 (p. 9): R(k^2 - 1) is at most every confluence width |W_(k,s)|Proposition 6.1 (p. 11): the transfer parity law A_k(r) = r + 1 mod 2 for r > 0 Proposition 7.1 (p. 12): |E_k| <= exp((2 log 2 + o(1)) log k / log log k) = k^(o(1))Proposition 7.5 (p. 13): |E_k| >= 2 always, |E_k| >= 3 for k = 3, 5 mod 8, and sum |E_k| >= (9/4)K + O(1)Theorem 10.3 (p. 32): every orbit of n + tau(n) reaches a value with more than G divisors within a primorial scale Theorem 10.6 (p. 33): two non-coalescing orbits of n + tau(n) have race gaps exceeding any G within a primorial scale Theorem 12.2 (p. 35): eventual two-branch collapse with opposite parity and infinitely many square gates implies coalescence Theorem 3.4 (p. 5): at most log X + 2 gamma + O(X^(-1/4)) components meet [1, X]Theorem 4.4 (p. 7): coalescence for n + tau(n) is equivalent to synchronization of the annular transfer maps Theorem 7.2 (p. 12): the exit sets E_k contain R elements of any residue class along a progression of k Theorem 8.14 (p. 23): conditionally on the unproved H_QE, sum of |E_k| is << K (log K)^2 Theorem 8.23 (p. 27): every fixed moment of |E_k| is << K (log K)^(C_m)Theorem 8.3 (p. 16): sum over 2 <= k <= K of |E_k| is << K (log K)^3, unconditionally Theorem 8.7 (p. 19): the shifted-square divisor estimate H_ST for tau(k^2 - j)^B over dyadic k

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