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Source. Conjecture 2.3, p. 5 of the author's manuscript, of Carl Pomerance, The first function and its iterates, in Connections in Discrete Mathematics, Cambridge University Press (2018), 125--138, as identified on the source card. Page numbers are those of the manuscript.

Statement

Here $s(n)=\sigma(n)-n$ and $s^{-1}(A)=\{n:s(n)\in A\}$.

Conjecture 2.3 (p. 5), quoted: "If $A$ is a set of natural numbers of asymptotic density 0, then $s^{-1}(A)$ has asymptotic density 0."

The paper introduces it with "In [9] the following conjecture is proposed" (p. 5), where [9] is Erdős, Granville, Pomerance and Spiro, On the normal behavior of the iterates of some arithmetic functions (1990). The paper does not prove it. In the proof of Theorem 2.4 (p. 6) it notes that the conjecture implies, by induction on $k$, that $s_k^{-1}(A)$ has density 0 for every $k\ge1$ when $A$ has density 0.

Proof pointer

None; it is a conjecture.

Dependencies

None. Read depth: claims checked; the statement was read on p. 5.

Bears on

• Problem 955: the conjecture is the problem's statement, with the same function $s$ and the same notion of density. The paper states it as a conjecture and proves nothing toward it; its Theorem 2.4 is conditional on it.