3. Let $\sigma(S)$ denote the sum of all elements in a non-empty set of integers $S$. Let $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of positive integers, and $a_{1}<a_{2}<\cdots<a_{n}$. For each positive integer $n \leqslant$ 1500, there exists a subset $S$ of $A$ such that $\sigma(S)=n$. Find the minimum value of $a_{10}$ that satisfies the above requirement.
Let $S_{k}=a_{1}+a_{2}+\cdots+a_{k} \quad(1 \leqslant k \leqslant 11)$. By the given conditions, there exists an index $m$ such that $S_{m-1}S_{k-1}+1$, and $a_{1}+\cdots+a_{k-1}=S_{k-1}$. Thus, there does not exist $S \simeq A$ such that $\sigma(S)=S_{k-1} +1$. Therefore,
$$
S_{k}=S_{k-1}+a_{k} \leqslant 2 S_{k-1}+1 .
$$

By the given conditions, $S_{1}=a_{1}=1$. Thus, by (1) and using induction, we can deduce that
$$
S_{k} \leqslant 2^{k}-1(1 \leqslant k \leqslant m) .
$$

From $1500 \leqslant S_{m} \leqslant 2^{m}-1$, we get $m=11$.
From (1), we have
$$
\frac{S_{k+1}-1}{2} \leqslant S_{k} \leqslant 2^{k}-1 \quad(k=1,2, \cdots, 10).
$$

Since $S_{k}$ is a natural number, we have
$$
\left(\frac{S_{k+1}-1}{2}\right) \leqslant S_{k} \leqslant 2^{k}-1 .
$$

Here $(x)$ denotes the smallest integer greater than or equal to $x$. In particular,
$$
S_{10} \geqslant\left(\frac{S_{11}-1}{2}\right) \geqslant\left(\frac{1500-1}{2}\right)=750 .
$$

Also, $S_{8} \leqslant 2^{8}-1=255$, so
$$
2 a_{10} \geqslant a_{9}+a_{10}=S_{10}-S_{8} \geqslant 495 .
$$

Therefore, $a_{10}=248$.
Next, we will use some of the above inequalities to construct a set $A$ that meets the requirements, and $a_{10}=248$. Specifically, we take
$$
\begin{array}{l}
A=\{1,2,4,8,16,32,64,128, \\
247,248,750\} .
\end{array}
$$

Since $255=2^{7}+2^{6}+\cdots+2+2^{0}$,
using binary representation, we can see that for $n \leqslant 255$, there exists $S \subset\{1,2,4, \cdots, 128\}$ such that $\sigma(S)=n$. Thus, for $n \leqslant 255+247=502$, there exists $S \subset\{1,2,4, \cdots, 128,247\}$ such that $\sigma(S)=n$; for $n \leqslant 502+248=750$, there exists $S \subset\{1,2, \cdots, 247,248\}$ such that $\sigma(S)=n$; for $n \leqslant 750+750=1500$, there exists $S \subset A$ such that $\sigma(S)=n$.