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Source. The unnumbered Conjecture at the end of Section 1, p. 2 of arXiv:1607.02863v2 (30 July 2024); the heuristic in Section 7, p. 6 (based on display (3), p. 4); the computation in Section 8, p. 7. Read on the PDF pages in the text layer. Preprint; a conjecture, not a result. Notation: $\tilde Q=\{n: d_n=D_n\}$, the complement of $\bigcup_{p\ge3}Q_p$, and $\tilde Q(x)=\#\{n\le x: n\in\tilde Q\}$.

Statement

Conjecture (p. 2). For some positive constants $K_1$ and $K_2$, every real $x>1$ satisfies

$$\frac{K_1x}{\log x}<\tilde Q(x)<\frac{K_2x}{\log x}.$$

The paper adds that "there are arbitrarily large $n$ with $d_n=D_n$" is itself unproved: "we have yet to discover why", and "we have not been able to emulate Euclid's elegant proof that there are infinitely many primes" (p. 6). The abstract states the infinitude as a conjecture.

Heuristic and evidence (the paper's, not a proof)

Display (3), p. 4: for a fixed odd prime $p$ and $x=p^b$, the count $Q_p(x)=|E_p|(p^b-p)/(p-1)$, printed as $\sim|E_p|x/p$ but asymptotic to $|E_p|x/(p-1)$ (see the Theorem 3 page), so along powers of $p$ a single prime removes a proportion about $|E_p|/(p-1)$ of the integers; the paper suggests that the set $\tilde Q$ "can be dealt with using methods applied to the set of primes" and bases the conjecture on (3) (Section 7, p. 6). Section 7 also describes a sieve of Eratosthenes-type listing of $\tilde Q$ through Theorem 2 (delete the intervals $mp^a\le n<(m+1)p^a$, $m\in E_p$), which avoids computing $H_n$. Section 8 reports 2641 values of $n\le10000$ with $d_n=D_n$, in 26 runs of consecutive integers, and tabulates how many odd primes below 10000 have each value of $|E_p|$. The count 2641 and the 26 runs were recomputed here by exact rational arithmetic and agree with one exception: the last run, printed as $9156_{155}$ (the paper's $a_b$ is the interval $a\le n<a+b$), has 156 members, $9156\le n\le9311$ ($q_{9311}=1$, $q_{9312}=97$), so the printed run lengths sum to 2640, not 2641. The recomputed values agree with the b-file of OEIS A110566.

Dependencies and read depth

A conjecture: nothing to depend on beyond Theorem 2 for the sieve and display (3) of the proof of Theorem 3 for the heuristic. Read depth: claims checked (statement read clause by clause; the computation of Section 8 recomputed to $x=10000$).

Bears on. #291: the conjecture is the quantitative form of the open half (infinitely many $n$ with $(a_n,L_n)=1$, of density zero) and the source of the site's heuristic $\asymp x/\log x$.