Given the problem:
- The terms form an arithmetic sequence where each term is $k$ greater than the previous, with $k$ an integer.
- The 16 th term is 81.
- A later term is 729 .
- We want to find how many distinct sequences satisfy this.

Step 1: Express the general term of the arithmetic sequence
For an arithmetic sequence with first term $\(a_1\)$ and common difference $k$, the $n$-th term is:

$$
\(a_n=a_1+(n-1) k\)
$$


Given:

$$
\(a_{16}=a_1+15 k=81\)
$$


Step 2: Express the later term
Let the later term be the $m$-th term with $\(m>16\)$ :

$$
\(a_m=a_1+(m-1) k=729\)
$$


Step 3: Use two equations to relate $\(a_1, k, m\)$
From the two equations:

$$
\(\begin{gathered}
a_1+15 k=81 \Longrightarrow a_1=81-15 k \\
a_1+(m-1) k=729
\end{gathered}\)
$$


Substitute $\(a_1\)$ :

$$
\(81-15 k+(m-1) k=729\)
$$


Simplify:

$$
\(\begin{gathered}
81+(m-16) k=729 \\
(m-16) k=648
\end{gathered}\)
$$


Step 4: Analyze the equation and conditions
- $m$ and $k$ are integers.
- $\(m>16\)$ (since the term is after 16 th).
- Since $m-16$ and $k$ are integers, and

$$
\((m-16) k=648\)
$$

we want to find all integer pairs $\((m-16, k)\)$ such that their product is 648 .
Step 5: Count the distinct integer pairs
Because the sequence terms and common difference must be integers, both $m-16$ and $k$ are integers that multiply to 648.
- $m-16$ must be a positive integer (since $m>16$ ), so $m-16>0$.
- $k$ can be positive or negative integers, but since $m-16>0$, to get a positive product ( $m-$ $16) k=648, k$ must also be positive.

Thus both $\(m-16\)$ and $k$ are positive integers.
Step 6: Find the positive divisors of 648
First, factor 648:

$$
\(648=2^3 \times 3^4\)
$$


Number of divisors:

$$
\((3+1) \times(4+1)=4 \times 5=20\)
$$


So 648 has 20 positive divisors.
Each divisor $d$ of 648 corresponds to one possible value of $m-16$, and uniquely determines $k=$ $\(\frac{648}{d}\)$.

Step 7: Determine possible values of $m$
For each divisor $d$ of 648, we have:

$$
\(m=d+16\)
$$


Since $m$ must be greater than 16 , all positive divisors $d \geq 1$ produce valid $m$.

Step 8: Check $\(a_1\)$ for each $k$
Recall:

$$
\(a_1=81-15 k\)
$$


Since $k$ is integer and $\(a_1\)$ can be any integer, $\(a_1\)$ is integer for all integer $k$.
No further restrictions.
Final:
Number of distinct sequences is the number of positive divisors of 648 , which is 20 .
Answer: There are 20 distinct sequences.