Erdos 1935 note sequences integers no one which

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theorem_p126: Erdős's 1935 theorem: for a sequence of integers no one of which divides another, the sum of 1/(a_n log a_n) converges, below a constant independent of the sequence, so every such sequence has lower density zero.

theorem_p127: Erdős's closing theorem of 1935: in Besicovitch's theorem lim inf can be replaced by lim, so for every epsilon > 0 and all large a the integers having a divisor between a and 2a have density less than epsilon.

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P. Erdős: Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), 126--128; Zentralblatt 12,52.

For a primitive sequence (A) = a_1, a_2, ... (a_m divides a_n only if m = n) the main Theorem (p. 126) is that sum_n 1/(a_n log a_n) converges; the paper derives it from the stronger inequality (1), sum_n (1/a_n) prod_{p <= p_n} (1 - 1/p) <= 1, where p_n is the largest prime factor of a_n, which gives the bound sum_n 1/(a_n log a_n) < c with c independent of the sequence. Erdős also records the easy fact that the upper density of a primitive sequence does not exceed 1/2 (no n+1 of its elements are at most 2n, since two would share the same odd part; a proof credited to M. Wachsberger and E. Weissfeld) and deduces from the Theorem that the lower density of every primitive sequence is zero (a footnote notes a different proof by Behrend). The paper sets these facts against the question of Chowla, Davenport and Erdős whether every primitive sequence has density zero, which Besicovitch had answered in the negative. The proof orders the a's by largest prime factor and counts multiples with the sieve of Eratosthenes. The closing section (stated p. 127, proved pp. 127-128) proves that the density of the integers with a divisor between a and 2a tends to 0, the source of that theorem for #446.

Result pages: theorem_p126 (the Theorem, inequality (1) and the density remarks, p. 126) and theorem_p127 (the closing theorem, p. 127, with its Lemma and the generalisation announced on p. 128). Both were read clause by clause against the page images of the print (claims checked).

Source: https://users.renyi.hu/~p_erdos/1935-04.pdf. No notice is printed in the file, an offprint scan whose first page (printed p. 126) carries only "[Extracted from the Journal of the London Mathematical Society, Vol. 10 (1935).]"; the society's journals page (https://www.lms.ac.uk/publications/jlms, read 2026-10-02) prints "© Copyright London Mathematical Society 2026", names Wiley as the publisher that handles rights and permissions through Wiley Online Library, and names no blanket license for the hybrid open-access journal, and Wiley Online Library could not be read on 2026-10-02; every other right reserved.

Bears on. #164: the Theorem (p. 126) shows that sum 1/(n log n) over a primitive set is finite, and inequality (1) bounds it by a constant independent of the set; it does not say which set maximises the sum. #143: for sets of integers greater than 1 the problem's hypothesis is primitivity, so the Theorem (p. 126) gives the problem's convergence assertion in that special case; it says nothing about sets containing non-integers. #892: the lower-density consequence (p. 126) shows that the problem's condition |A cap [1,2^{n_i}]| >> 2^{n_i} cannot hold when the n_i run over all large integers, a necessary condition only. #446: the closing theorem (p. 127) gives delta(n) -> 0, with no rate, and does not touch the question on delta_1(n).

Results to transcribe.

• Theorem (p. 126): For any primitive sequence a_1, a_2, ... (a_m divides a_n only if m = n), the series sum_n 1/(a_n log a_n) converges; inequality (1) gives the bound c, a constant independent of the sequence.
• Inequality (1), p. 126: sum_{n} (1/a_n) prod_{p <= p_n} (1 - 1/p) <= 1, where p_n is the greatest prime factor of a_n.
• Upper density, p. 126: The upper density of a primitive sequence does not exceed 1/2: it cannot contain n+1 elements at most equal to 2n.
• Lower density: Every primitive sequence has lower density zero (a consequence of the Theorem, p. 126).
• Closing theorem, p. 127 (proof pp. 127-128): in Besicovitch's theorem lim inf can be replaced by lim. For every epsilon > 0 and a > a(epsilon), the density of the integers having a divisor between a and 2a is less than epsilon. The proof uses a lemma, stated without proof as easily proved by Turán's method, that the normal number of prime factors less than a of an integer is log log a. The note ends by announcing, without proof, that the density of the integers with a divisor between n and n^{1+epsilon_n} tends to 0 as n -> infinity when epsilon_n -> 0.

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