How to find values that occur specific number of times in Pandas Series?

Your logic is fine, you just shouldn't repeat the most expensive part, which is the counting. Store this in a variable and reuse. You may also not need to retrieve the underlying NumPy array, pd.Index objects are often sufficient:

sr = pd.Series([5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8])

counts = sr.value_counts()

res = counts[counts == 3].index
# Int64Index([8, 6, 5], dtype='int64')

The reason there's no ready-made method for what you want is any solution will require minimum O(n) time complexity, which is the complexity for value_counts. There's no way round this.

One alternative, dict-based collections.Counter, will be less efficient when it comes to filtering by count. Since NumPy arrays are stored efficiently in memory, Boolean filtering is efficient relative to dictionary iteration.

@jpp's answer is probably the one you should go with, but here is a weird alternative (just for fun):

sr.groupby(sr).filter(lambda x: len(x) == 3).unique()
#array([5, 6, 8])

Using loc

sr.value_counts().loc[lambda x : x==3].index
Out[162]: Int64Index([8, 6, 5], dtype='int64')

You could also use .where:

sr.where(sr.value_counts()==3).dropna().index

# Output:
Int64Index([5, 6, 8], dtype='int64')