1. We need to count functions

$f:A\to P(A)$

where

$A=\{1,2,3,\dots,7\}$

and the condition is

$a\in f(a),\quad \forall a\in A.$

2. Understand the condition for each element

For each fixed $a\in A$, the value $f(a)$ is a subset of $A$ such that $a$ must belong to that subset.

So, to form $f(a)$:

• $a$ is compulsory,
• each of the remaining $6$ elements may be either included or not included independently.

Hence, the number of possible choices for $f(a)$ is

$2^6=64.$

3. Count all such functions

There are $7$ elements in $A$, and for each $a\in A$, there are $64$ choices for $f(a)$. These choices are independent.

Therefore, total number of functions is

$64^7=(2^6)^7=2^{42}.$

4. Write in the form $m^n$ with least $m$

We need

$2^{42}=m^n$

with $m,n\in \mathbb N$ and $m$ least.

To make $m$ least, we take the smallest possible integer base, namely

$m=2,\quad n=42.$

5. Compute $m+n$

$m+n=2+42=44.$

So the required answer is

$\boxed{44}$