#### References to other Issues or PRs

Issue: https://github.com/sympy/sympy/issues/28806

#### Brief description of what is fixed or changed

Adds the missing type annotations in `sympy/ntheory/partitions_.py`:

- `_pre() -> None`, with module-level declarations `_factor: list[int]` and `_totient: list[int]` (annotation only, so the `'_factor' not in globals()` check in `_partition` keeps working)
- `_a(n: int, k: int, prec: int) -> MPF_TUP`
- `_d(n: int, j: int, prec: int, sq23pi: MPF_TUP, sqrt8: MPF_TUP) -> MPF_TUP`; the local rebinding `j = from_int(j)` was renamed to `jm` so the parameter keeps a single type
- `_partition_rec`: `prev: list[int]`
- inner `_M(n: int, N: int) -> float`
- `npartitions(n: int, verbose: bool = False) -> Integer`

`MPF_TUP` is the existing alias from `sympy.core.evalf` (raw mpmath mpf tuple), imported under `TYPE_CHECKING`.

Verified locally: `mypy sympy` reports no new errors (only the 2 pre-existing `no-redef` errors in `sympy/__init__.py`, also present on master), `pytest sympy/ntheory/tests/test_partitions.py` passes, and `_partition(100) == 190569292`.

#### Other comments

Typing-only change, no behavior change intended.

#### AI Generation Disclosure

This PR (code and description) was prepared with Claude Code (Anthropic), operated by me. All changed lines in `sympy/ntheory/partitions_.py` were AI-generated and then verified locally with mypy and the test suite as described above.

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