A friend of mine taught me this question. I found that it is related with 'Postage stamp problem' (though it does not seem to be same).

Let $m,a_1\lt a_2\lt \cdots\lt a_n$ be natural numbers. Now let us consider the following condition :

**Condition** : For each $k\in\mathbb N$ which satisfies $1\le k\le m$, $k$ can be represented as $a_i, 2a_i$ or $a_i+a_j\ (i\not=j)$.

Letting $\min_m(n)$ be the min of $n$, then here are my questions.

Question 1: What is $\min_{m=100}(n)$ under the condition?

Question 2: What is $\min_{m}(n)$ under the condition?

**Example** : For $m=100$, $(1,2,\cdots, 8,9,10, 20, \cdots,80,90)$ is an obvious example with $n=18$.

**Motivation** : For $m=100$, it was almost by chance that I discovered the following example with $n=16$ :

$$(1,3,4,7,8,9,16,17,21,24,35,46,57,68,79,90)$$ I expect that $\min_{m=100}(n)$ would be $16$, but I can't prove it.

For smaller $m$, we can get $\min_{m}(n)$ such as $$\min_{m=1}(n)=\min_{m=2}(n)=1,\min_{m=3}(n)=\min_{m=4}(n)=2,$$$$\min_{m=5}(n)=\min_{m=6}(n)=\min_{m=7}(n)=\min_{m=8}(n)=3,\min_{m=9}(n)=4.$$

Since we know that $(1,2,\cdots,k)$ is an example for $m=2k$, we get $\min_{m=2k}(n)\le k$ for any $k\in\mathbb N$. However, I don't have any good idea. Can anyone help?

*Update* : For $m=100$, I found an interesting example with $n=16$ :
$$(1,2,3,7,11,15,19,23,27,28,29,30,61,64,67,70)$$
Using arithmetic progressions might be a key.