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Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference [or a proof!] for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.

Erdös and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.

Erdös and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find it in that 400-page book (called "Sieve Methods") but the version I found online is not searchable, so I thought I'd ask if maybe somebody knows a precise reference for this result?

For every $\epsilon'>0$, there is a constant $c=c(\epsilon')$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon'}$.

Erdös and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.