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Source. Theorem 1, Section 1, p. 2 of arXiv:1607.02863v2 (30 July 2024; the paper is dated 29 July 2024), proof in Section 2, pp. 3--4; read on the PDF pages in the text layer. Preprint, not published in a journal (arXiv listing checked). Notation: $H_n=c_n/d_n$ in lowest terms and $D_n=\mathrm{lcm}(1,2,\ldots,n)$.

Statement

Theorem 1 (p. 2). "For all $n>1$, we have $c_n\ne c_{n-1}$. Also, each of the following holds for infinitely many $n$:

(i) $d_n>d_{n-1}$, (ii) $d_n=d_{n-1}$, (iii) $d_n<d_{n-1}$; (iv) $c_n>c_{n-1}$, (v) $c_n<c_{n-1}$."

Proof pointer and sketch (Section 2)

(i) $p\mid d_p$ for every prime $p$, so $d_n$ is unbounded. (ii) For $n=2p>6$, $p$ divides both $d_{n-1}$ and $d_n$, and the two-term relations between $c_n/d_n$ and $c_{n-1}/d_{n-1}$ give $d_n\mid d_{n-1}$ and $d_{n-1}\mid d_n$. (iii) For $n=p(p-1)$, $H_{n-1}=H_{p-2}/p+S_0+\cdots+S_{p-2}$ with $S_i=\sum_{j=1}^{p-1}1/(ip+j)$: the $p-2$ terms whose denominators are multiples of $p$ sum to $H_{p-2}/p$, and each block $S_i$ of the other terms has a reduced numerator divisible by $p$; since $p\mid c_{p-1}$ (the pairing $1/j+1/(p-j)$, display (1)), $p\nmid d_n$ while $p\mid d_{n-1}$, and $d_n\le nd_{n-1}/p^2<d_{n-1}$. (iv) follows from (i); (v) from (iii) by a short computation. The final paragraph shows $c_n=c_{n-1}$ would force $d_{n-1}\ge3d_n$ and $H_n\ge3H_{n-1}$, impossible. The proofs are elementary and complete on pp. 3--4; they were read through here but not independently reviewed.

Dependencies and read depth

Elementary; the paper notes that Wolstenholme's theorem is not needed. Read depth: claims checked; the proof read through, not verified.

Relation to Problem 291

Part (iii) gives infinitely many $n$ with $d_n<d_{n-1}$, and any such $n$ has $d_n<D_n$, that is $(a_n,L_n)>1$ in the notation of Problem 291; this is a second route to the trivial half of that problem, beside the leading-digit observation the site records. The theorem says nothing about the open half, $d_n=D_n$ infinitely often, which the paper states as a conjecture (the conjecture page). The non-monotonicity of $d_n$ is also the $a=1$ case of the denominator question of Problem 290, treated on that problem's page.

Bears on. #291 (part (iii) as a route to the trivial half); #290 (part (iii) answers the existence question for $a=1$).