Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 446

claims/: The 1 claim page of Problem 446, one per claimant's result; the problem's standing derives from them.

Statement. Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$ . What is the growth rate of $\delta(n)$ ?

If $\delta_1(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta_1(n)=o(\delta(n))$ ?

Status. Solved. The site's label; Ford determined the order of $\delta(n)$ and answered the second question no, as the claim page below records.

Source. erdosproblems.com/446, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #446, https://www.erdosproblems.com/446.

References.

[Be34] Besicovitch, A., On the density of certain sequences of integers. Math. Annalen (1934), 336-341.
[Er35] Erdős, Paul, Note on Sequences of Integers No One of Which is Divisible By Any Other. J. London Math. Soc. (1935), 126-128.
[Er60] Erdős, P., An asymptotic inequality in the theory of numbers. Vestnik Leningrad. Univ. (1960), 41-49.
[Fo08] Ford, Kevin, The distribution of integers with a divisor in a given interval. Ann. of Math. (2) (2008), 367-433.
[Te84] Tenenbaum, G., Sur la probabilité qu'un entier posséde un diviseur dans un intervalle donné. Compositio Math. (1984), 243-263.

Formalization. None recorded.

Current assessment

The question is the site's formulation, accessed and unchanged, in two parts: the growth rate of $\delta(n)$ , the density of integers with a divisor in $(n,2n)$ , and whether $\delta_1(n)$ , the density of those with exactly one such divisor, is $o(\delta(n))$ . Both parts are settled by Ford [Fo08]: the first by the order of magnitude displayed on the claim page, the second in the negative.

The first part has a long history. Besicovitch [Be34] showed $\liminf\delta(n)=0$ , which gives a primitive set of positive upper density; Erdős [Er35] showed $\delta(n)\to0$ (card); Erdős [Er60] found the exponent, $\delta(n)=(\log n)^{-\alpha+o(1)}$ with $\alpha=1-(1+\log\log2)/\log2=0.08607\ldots$ ; Tenenbaum [Te84], Theorem 1, pinned the count of integers with a divisor in $[y,z]$ between bounds that match up to slowly varying factors (card); and Ford [Fo08], Corollary 2, gave the exact order $\delta(n)\asymp(\log n)^{-\alpha}(\log\log n)^{-3/2}$ (card). For the second part, the site records that Erdős raised it in his Oberwolfach problem collection, expecting $\delta_1(n)=o(\delta(n))$ while noting that Tenenbaum's results told against it. Ford's Theorem 4 and Corollary 7 give $\delta_r(n)\gg_r\delta(n)$ for every fixed $r\ge1$ , where $\delta_r$ is the density of integers with exactly $r$ divisors in $(n,2n)$ , so the expectation fails already at $r=1$ . The claim page Ford 2008 records both results, the refereed venue and the curator's credit, and the problem's standing derives from it.

Nothing in the two questions remains open. The dyadic variant with the divisor count itself is Problem 448, and Problem 692 and Problem 693 ask related questions about divisors in short intervals. No formalization is recorded, and this repository has not checked Ford's proof independently; the account rests on the site page, Ford's paper and the cards above.

Linked library material

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Linked from (18)

Ford: the order of delta(n), and delta_1(n) is not o(delta(n))problems/divisors/E0446/claims Problem 448 Divisors and Multiples Divisors and Multiples divisors/erdos_1935_note_sequences_integers_no_one_which Theorem (p. 127): the density of integers with a divisor between a and 2a tends to zero divisors/ford_2008_distribution_integers_divisor_given_interval Corollary 2 (p. 372): H(x,y,cy) and epsilon(y,cy) have order 1/((log)^delta (loglog)^(3/2))Corollary 7 (p. 376): epsilon_r(y,lambda y) >> epsilon(y,lambda y), while epsilon_r(y,z) = o(epsilon(y,z)) as z/y grows Theorem 4 (p. 375): H_1(x,y,z)/H(x,y,z) has order log log(z/y+10)/log(z/y+10)Theorem 5 (p. 376): bounds for H_r(x,y,z)/H(x,y,z) when r >= 2 divisors/tenenbaum_1984_sur_la_probabilite_qu_un Open problem (p. 246): is epsilon'(y)/epsilon(y) = o(1), for the densities of integers with exactly one and with at least one divisor in [y, 2y)?Theorem 1 (pp. 243-244): H(x, y, z) lies between x u^delta L_1(1/u) and x u^delta L_2(1/u) when 1 < 2y <= z <= min(y^{3/2}, x^{1/2})Theorem 2 (p. 250): H(x, y, (1+eta)y) for 0 < eta <= 1, asymptotic to eta x under (*) and equal to x (log y)^{-A((1+gamma)/log 2)+o(1)} when gamma <= log 4 - 1 + o(1)On the density of certain sequences of integers Theorem 1 (p. 339): the dyadic divisor-window densities have average zero

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