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Statement

Notation as on the Proposition 1 page: $\mathrm{Pow}(A;s)$ is the nondecreasing sequence of the powers $a^k$ with $a\in A$ and $k\ge s$, and $\Sigma(S)$ is the set of finite subset sums of a sequence $S$.

Conjecture 1 (p. 257, quoted). "Let $s\geq1$ and let $A$ be a sequence of integers $\geq2$. If for every $a_1,a_2\in A$, $\gcd\{a_1,a_2\}=1$ and $\sum_{a\in A}\frac{1}{\log a}>\log 2$ then $\Sigma(\mathrm{Pow}(A;s))$ has a positive lower asymptotic density."

The paper introduces it as a generalization of Erdős's question, which it reports just before, asking for a proof that $n_k\ll k$, where $n_1<n_2<\cdots$ are the positive integers that are sums of distinct powers of $3$ and of $4$; it reports $n_k\ll k^{1.0353}$ as the best known result, from its reference [11] (G. Melfi, An additive problem about powers of fixed integers, Rend. Circ. Mat. Palermo (II) 50 (2001), 239--246).

Remark after the conjecture (p. 257). The paper states that the conjecture becomes false if pairwise coprimality is weakened to $\gcd\{a\in A\}=1$: for $A=\{3,9,81,104\}$ one has $\gcd\{a\in A\}=1$ and $\sum_{a\in A}1/\log a>\log2$, while $\Sigma(\mathrm{Pow}(A;s))$ has lower asymptotic density zero, citing [11].

The printed threshold (observations of this page, not of the paper). The quantifier "for every $a_1,a_2\in A$" must be read for distinct $a_1,a_2$, since $\gcd\{a,a\}=a\ge2$. With that reading, the single-element set $A=\{3\}$ satisfies the printed hypothesis, as $1/\log3\approx0.910>\log2$ for the natural logarithm, yet $\Sigma(\mathrm{Pow}(\{3\};s))$ consists of integers with base-$3$ digits $0$ and $1$ and has density zero. So the conjecture as printed fails, and the intended threshold is presumably $1/\log2$, the threshold of the related question of Burr, Erdős, Graham and Li recorded on the Problem 125 page. The example $\{3,9,81,104\}$ of the remark has $\sum1/\log a\approx1.81$, above both thresholds.

Source. Conjecture 1 and the remark after it, p. 257, of Giuseppe Melfi, On certain positive integer sequences, Riv. Mat. Univ. Parma (7) 3* (2004), 253--260, as identified on the source card.

Read depth. Claims checked: the conjecture, the remark and the preceding report of Erdős's question were read clause by clause on p. 257. The density claim for $\{3,9,81,104\}$ is cited by the paper to [11], which is not held here, and is not checked. Nothing here is independently reviewed.

Proof pointer

None: the paper poses the statement as a conjecture.

Dependencies

The remark rests on [11], cited above.

Bears on

• Problem 125: the problem asks whether $A+B$ has positive lower density, where $A$ and $B$ are the integers with only the digits $0,1$ in base $3$ and in base $4$. For the bases $\{3,4\}$, which are coprime with $1/\log3+1/\log4\approx1.63$, above both $\log2$ and $1/\log2$, the conjecture asserts positive lower density for $\Sigma(\mathrm{Pow}(\{3,4\};s))$, a subset of $A+B$ for every $s\ge1$ (an observation of this page), so it would answer Problem 125 yes. The Problem 125 page records a disproof, lower density zero for $A+B$, given in Lean with no refereed publication; if that result holds, Conjecture 1 fails at $\{3,4\}$ under either threshold.