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Claim. Let $s_1<s_2<\cdots$ be the squarefree numbers. For every real $0\le\alpha\le3$ there is a constant $B(\alpha)>0$ with $\sum_{s_{n+1}\le x}(s_{n+1}-s_n)^\alpha\sim B(\alpha)\,x$, so the limit that Problem 145 asks about exists for every $\alpha$ in that range. The source is C. Hooley, On the distribution of square-free numbers, Canad. J. Math. 25 (1973), no. 6, 1216--1223, whose abstract states that Erdős's formula, known for $0\le\gamma\le2$, holds for the wider range $0\le\gamma\le3$, by a method different from the one that gave similar results for the sums of two squares and for the integers coprime to a given large modulus. Chan's 2023 paper, which extends the range further, cites this paper for $0\le\gamma\le3$.

The site's reference key [Ho73] prints a different paper of Hooley's, On the intervals between consecutive terms of sequences, Proc. Sympos. Pure Math. 24 (1973), 129--140, doi:10.1090/pspum/024/0384742; the moment result for squarefree gaps that the site's commentary credits to Hooley is the Canadian Journal paper cited above, and the problem page's reference entry records the collision.

Covers. Every exponent $0\le\alpha\le3$; the exponents $0\le\alpha\le2$ are already covered by Erdős 1951, and the exponents $3<\alpha<3.75$ by Huxley 1997 and Chan 2023.

Acceptance. Refereed: Canadian Journal of Mathematics, volume 25, issue 6 (December 1973), pp. 1216--1223; the Crossref record of the DOI gives these data. The site labels the problem OPEN (page last edited 19 October 2025), so its curator's remark that Hooley extended the result to $\alpha\le3$ is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The corpus holds no card for this paper, has not reproved the theorem and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper.