Problem 945

Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the divisors of $m$ ). Estimate $F(x)$ . In particular, is it true that

F(x) \leq (\log x)^{O(1)}?

In other words, is there a constant $C>0$ such that, for all large $x$ , every interval $[x,x+(\log x)^C]$ contains two integers with the same number of divisors?

Source: erdosproblems.com/945

Number theory
Divisors

Wiki page

Status

Open

No claim settles this problem.

Er85e: Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
ErMi52: Erdős, P. and Mirsky, L., The distribution of values of the divisor function $d(n)$ . Proc. London Math. Soc. (3) (1952), 257-271.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0; section B18 "Solutions of $d(n)=d(n+1)$ ", printed p. 112, where the book reports the Erdős and Mirsky question with the guess $k=(\ln n)^c$ . Library home: Guy 2004.

Problem 945

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Proof claims

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