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I am interested in the following problem. An arbitrary integer $n \geq 2$ is given. Find the minimum integer $m \geq 1$ such that there exist integers $0\lt a_1\lt a_2\lt \cdots \lt a_m$ satisfying the following property: for every integer $k \in \{2,\ldots,n\}$ there exist $i,j \in \{1,\ldots,m\}$ such that $k = a_i + a_j$.

To illustrate, let $n=5$. We are forced to take $a_1=1$, so that $2=a_1+a_1$. Then, we are forced to take $a_2=2$, so that $3 = a_1+a_2$. We get $4 = a_2+a_2$ ``for free''. Finally, we need to take $a_3=3$ to get $5=a_2+a_3$ or $a_3=4$ to get $5=a_1+a_3$. In any case, we see that $n=5$ corresponds to $m=3$.

My question: what is $m$ for arbitrary $n$? And if you do not know the answer: can you provide a non-trivial lower bound for $m$? Or do you know a problem which is equivalent or very similar to this one?

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  • $\begingroup$ What do you call nontrivial? There are only $(m+1)$-choose-$2$ pairs selected from $m$ numbers, so a lower bound on $m$ is given by ${m+1\choose2}\ge n-1$. $\endgroup$ – Gerry Myerson Jul 21 at 2:53
  • $\begingroup$ Is it oeis.org/A066063 $\endgroup$ – Gerry Myerson Jul 21 at 3:01
  • $\begingroup$ Are you still here, Berry? $\endgroup$ – Gerry Myerson Jul 22 at 13:16
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One strategy is to have your set $\{1,2,3,\ldots k,2k,3k,4k\ldots\}$ If you go up to $k^2$ you can get all sums up to $k^2+k$ with $2k$ numbers in the set. It means for numbers up to $n$ you choose $k$ so that $$k^2+k=n\\k=\frac 12(-1+\sqrt{1+4n})\approx \sqrt n\\m \approx 2\sqrt n$$ I don't know if this is optimal, but it is a target for others to shoot at.

Added: For a lower bound, note that there are at most $\frac 12m(m+1)$ sums possible. We then need $$\frac 12m(m+1) \ge n\\ (m+\frac 12)^2 \ge 2n+\frac 14\\ m \ge \sqrt{2n+\frac 14}-\frac 12$$ which shows the above is close. It is a factor $\sqrt 2$ higher because some of the sums match.

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  • $\begingroup$ Nice observation, but indeed not what I'm looking for. I suppose your answer bounds $m$ from above, while (if $m$ cannot be found exactly) I'm only interested in (non-trivial) lower bounds for $m$. I updated my question accordingly. $\endgroup$ – Berry van Peer Jul 20 at 15:07
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This is one variant of the postage stamp problem. OEIS sequence A001212 gives several values and references.

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  • $\begingroup$ That sequence is not the right variant though. $\endgroup$ – orlp Jul 21 at 0:46
  • $\begingroup$ OK, I guess the difference is that you require exactly (instead of at most) two positive summands. $\endgroup$ – Rob Pratt Jul 21 at 0:52

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