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The claim. Let $s_1<s_2<\cdots$ be the squarefree numbers, the set $B$ of Problem 489 for $A=\{p^2:p\text{ prime}\}$, and for $t\ge1$ let $\beta_t$ be the density of the $s_i$ with $s_{i+1}-s_i=t$, which exists (Lemma 1, p. 107). Then

$$\sum_{s_{i+1}\le x}(s_{i+1}-s_i)^2=x\sum_{t\ge1}t^2\beta_t+o(x), \qquad\sum_{t\ge1}t^2\beta_t<\infty,$$

so the mean squared gap tends to a finite limit. This is the case $\alpha=2$ of display (23), p. 107, of P. Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109, received 4 May 1951 (the date this page is named by). The paper conjectures (23), $\sum_{s_{i+1}\le x}(s_{i+1}-s_i)^\alpha=C_\alpha x+o(x)$, for every $\alpha$, says that it can be proved for $\alpha<A$ with $A$ between $2$ and $3$, and sketches the proof for $\alpha=2$: Lemma 2 (p. 107) bounds the number of $s_i<x$ with gap above $t$ by $c_{17}x/(t^2(\log t)^2)$, because a gap of length $r$ contains at least $r/16$ integers divisible by the square of a prime above $t\log t/100$, by the Chebyshev prime count and the convergence of $\sum1/P^2$; displays (28) and (29), pp. 108--109, then compare the sum over large gaps with Lemma 1's densities. The card Erdős (1951) records the result. Erdős's 1961 problem list, the site's source for the problem (Erdős (1961), printed pp. 236--237), states the result as (I.28.2), with the normalization $\frac1n\sum_{s_i<n}$, cites this paper for it, and poses the general question as (I.28.3) for a sequence with $a_k/k^2\to\infty$, adding that under $a_k<ck^2$ alone the limit need not exist. The paper's closing remarks (p. 109) bear on the general case: for $a_1<a_2<\cdots$ with $\sum1/a_i<\infty$ the gap densities of the non-multiples exist, and for every $\varepsilon$ the gaps above a constant $c_\varepsilon$ contribute less than $\varepsilon x$ to the first moment below $x$, but for the $a$'s in the intervals $[2^k,2^k(1+1/k^2)]$ the normalized $(1+\varepsilon)$-moment of the gaps is unbounded although $\sum1/a_i<\infty$; whether this can happen for pairwise coprime $a$'s is left open there.

Covers. The instance $A=\{p^2:p\text{ prime}\}$, for which $B$ is the squarefree numbers: the limit exists and is finite. Not covered: any other $A$.

Acceptance. Refereed: the journal publication cited above. The site's commentary credits Erdős with the existence of the limit in this case; on a problem the site labels OPEN, that credit is context, not acceptance.

Depends on. Nothing in this wiki.