Determine the range of the following function defined for  integer $k$,

\[f(k)=(k)_3+(2k)_5+(3k)_7-6k\]

where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$
To determine the range of the function \( f(k) = (k)_3 + (2k)_5 + (3k)_7 - 6k \), we need to understand the behavior of each term in the function. Here, \((k)_{2n+1}\) denotes the multiple of \(2n+1\) closest to \(k\).

1. **Analyzing \((k)_3\):**
   \[
   (k)_3 \text{ is the multiple of } 3 \text{ closest to } k.
   \]
   This means:
   \[
   |k - (k)_3| \leq 1
   \]
   For example, if \(k = 4\), then \((4)_3 = 3\) because 3 is the closest multiple of 3 to 4.

2. **Analyzing \((2k)_5\):**
   \[
   (2k)_5 \text{ is the multiple of } 5 \text{ closest to } 2k.
   \]
   This means:
   \[
   |2k - (2k)_5| \leq 2
   \]
   For example, if \(k = 3\), then \(2k = 6\) and \((6)_5 = 5\) because 5 is the closest multiple of 5 to 6.

3. **Analyzing \((3k)_7\):**
   \[
   (3k)_7 \text{ is the multiple of } 7 \text{ closest to } 3k.
   \]
   This means:
   \[
   |3k - (3k)_7| \leq 3
   \]
   For example, if \(k = 2\), then \(3k = 6\) and \((6)_7 = 7\) because 7 is the closest multiple of 7 to 6.

4. **Combining the terms:**
   \[
   f(k) = (k)_3 + (2k)_5 + (3k)_7 - 6k
   \]
   Using the bounds derived:
   \[
   |k - (k)_3| \leq 1, \quad |2k - (2k)_5| \leq 2, \quad |3k - (3k)_7| \leq 3
   \]
   Therefore:
   \[
   -1 \leq k - (k)_3 \leq 1, \quad -2 \leq 2k - (2k)_5 \leq 2, \quad -3 \leq 3k - (3k)_7 \leq 3
   \]
   Summing these inequalities:
   \[
   -1 - 2 - 3 \leq k - (k)_3 + 2k - (2k)_5 + 3k - (3k)_7 \leq 1 + 2 + 3
   \]
   Simplifying:
   \[
   -6 \leq k + 2k + 3k - ((k)_3 + (2k)_5 + (3k)_7) \leq 6
   \]
   \[
   -6 \leq 6k - ((k)_3 + (2k)_5 + (3k)_7) \leq 6
   \]
   Rearranging:
   \[
   -6 \leq -f(k) \leq 6
   \]
   Multiplying by -1:
   \[
   -6 \leq f(k) \leq 6
   \]

5. **Verifying the range:**
   We need to check if each integer from -6 to 6 can be achieved by some integer \(k\). By testing specific values of \(k\), we can confirm that each integer in the range \(\{-6, -5, \ldots, 6\}\) can indeed be obtained.

Conclusion:
The range of the function \( f(k) \) is \(\{-6, -5, \ldots, 6\}\).

The final answer is \( \boxed{ \{-6, -5, \ldots, 6\} } \)