Erdős Problem #347

Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\]has density $1$ for every cofinite subsequence $A'$ of $A$?

#347: [ErGr80]

number theory | complete sequences

This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and van Doorn, also in the comments).

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This page was last edited 22 January 2026.

External data from the database - you can help update this
Formalised statement? Yes

17 comments on this problem

Likes this problem	Woett, ebarschkis, jbbaehr22
Interested in collaborating	None
Currently working on this problem	jbbaehr22
This problem looks difficult	None
This problem looks tractable	ebarschkis
The results on this problem could be formalisable	None
I am working on formalising the results on this problem	None

Additional thanks to: ebarschkis, Terence Tao, and Wouter van Doorn

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #347, https://www.erdosproblems.com/347, accessed 2026-02-13