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Updated Oct 8

Problem 839

Statement. Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$ . Is it true that

\limsup \frac{a_n}{n}=\infty?

Or even

\lim \frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?

Status. Open; the site labels the problem OPEN.

Source. erdosproblems.com/839, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #839, https://www.erdosproblems.com/839.

References.

[Fr93] R. Freud, Adding numbers - on a problem of P. Erdős. James Cook Mathematical Notes (1993), 6199-6202 (the site's key; the issue prints the title "Adding numbers").

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (16)

Sequences and Densities of Integers additive_combinatorics/freud_1993_adding_numbers_problem_p The construction of pp. 6199–6201: a set of 76y − 7 integers up to 144y − 12 in which no member is a sum of two or more consecutive members, density 19/36 Sequences and Densities of Integers integer_sequences/erdos_1977_problems_results_combinatorial_number_theory_iii Erdős: Many old and on some new problems of mine in number theory Problem III.3 (pp. 20–21): density questions for sequences with no term a sum of consecutive terms, a construction of upper density 1/2, and the bound (1)integer_sequences/erdos_1992_my_forgotten_problems_number_theory number_theory/erdos_1980_old_new_problems_results_combinatorial_number_theory number_theory/erdos_1982_some_new_problems_results_number_theory Construction (pp. 56–57): a sequence with no term a sum of consecutive earlier terms and upper density 1/2 number_theory/guy_1991_western_number_theory_problems Problem 91:02 (p. 9): must n+2 integers up to 2n contain one that is a sum of consecutive members? Pomerance's counterexample ramsey_theory/erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk Question (p. 160, unnumbered): if no m_n is a sum of consecutive terms, has the sequence lower density 0, or even logarithmic density 0?Theorem (pp. 160--161, printed TEOREM, unnumbered): if all consecutive sums of A are distinct, a_n > cn log n infinitely often and the reciprocals of A in (u, u^2) sum to less than C

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