REF: use dtype.freq less in PeriodDtype (#52585)

* REF: use dtype.freq less in PeriodDtype

* mypy fixup

Author: jbrockmendel

pandas/_libs/tslibs/dtypes.pyx

@@ -53,7 +53,10 @@ cdef class PeriodDtypeBase:
     @property
     def _freqstr(self) -> str:
         # Will be passed to to_offset in Period._maybe_convert_freq
-        return _reverse_period_code_map.get(self._dtype_code)
+        out = _reverse_period_code_map.get(self._dtype_code)
+        if self._n == 1:
+            return out
+        return str(self._n) + out
 
     cpdef int _get_to_timestamp_base(self):
         """

pandas/_libs/tslibs/period.pyx

@@ -1688,6 +1688,8 @@ cdef class _Period(PeriodMixin):
             # We already have a dtype code
             dtype = PeriodDtypeBase(freq, 1)
             freq = dtype._freqstr
+        elif isinstance(freq, PeriodDtypeBase):
+            freq = freq._freqstr
 
         freq = to_offset(freq)
 
@@ -2372,7 +2374,7 @@ cdef class _Period(PeriodMixin):
         """
         Return a string representation of the frequency.
         """
-        return self.freq.freqstr
+        return self._dtype._freqstr
 
     def __repr__(self) -> str:
         base = self._dtype._dtype_code

pandas/core/dtypes/dtypes.py

@@ -863,6 +863,7 @@ class PeriodDtype(PeriodDtypeBase, PandasExtensionDtype):
     _match = re.compile(r"(P|p)eriod\[(?P<freq>.+)\]")
     _cache_dtypes: dict[str_type, PandasExtensionDtype] = {}
     __hash__ = PeriodDtypeBase.__hash__
+    _freq: BaseOffset
 
     def __new__(cls, freq):
         """
@@ -886,7 +887,7 @@ def __new__(cls, freq):
             return u
 
     def __reduce__(self):
-        return type(self), (self.freq,)
+        return type(self), (self.name,)
 
     @property
     def freq(self):
@@ -940,7 +941,7 @@ def __str__(self) -> str_type:
 
     @property
     def name(self) -> str_type:
-        return f"period[{self.freq.freqstr}]"
+        return f"period[{self._freqstr}]"
 
     @property
     def na_value(self) -> NaTType:
@@ -955,12 +956,6 @@ def __eq__(self, other: Any) -> bool:
     def __ne__(self, other: Any) -> bool:
         return not self.__eq__(other)
 
-    def __setstate__(self, state) -> None:
-        # for pickle compat. __getstate__ is defined in the
-        # PandasExtensionDtype superclass and uses the public properties to
-        # pickle -> need to set the settable private ones here (see GH26067)
-        self._freq = state["freq"]
-
     @classmethod
     def is_dtype(cls, dtype: object) -> bool:
         """