## Task

Given a positive integer, `n`, return the number of possible ways such that `k` positive integers multiply to `n`. Order matters.

**Examples**
```
n = 24
k = 2
(1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), (24, 1) -> 8

n = 100
k = 1
100 -> 1

n = 20
k = 3
(1, 1, 20), (1, 2, 10), (1, 4, 5), (1, 5, 4), (1, 10, 2), (1, 20, 1),
(2, 1, 10), (2, 2, 5), (2, 5, 2), (2, 10, 1), (4, 1, 5), (4, 5, 1),
(5, 1, 4), (5, 2, 2), (5, 4, 1), (10, 1, 2), (10, 2, 1), (20, 1, 1) -> 18
```
**Constraints**
`1 <= n <= 500_000_000`
and `1 <= k <= 1000`

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "# I'm slightly sad, I had a simple clean solution, that wasn't quite fast enough, needed 2-3 more seconds\n#\n# def multiply(n, k):\n#     return k if k<2 else sum(multiply(x,k-1) for x in factors(n))\n#\n\nfrom collections import Counter\nfrom functools import lru_cache, reduce\nfrom itertools import combinations\nfrom operator import mul\nfrom math import factorial\n\ndef multiply(n, k):\n    if k<2: return k\n    if k==2: return len(factors(n))\n    #get all the combinations that prod to n\n    tups = get_factor_tuples(n,k)\n    #insert all the extra 1s\n    tups = { tuple((1,)*(k-len(x))+x) for x in tups }\n    #do factorial math to find # of unique combinations of each tuple (multinomial coefficient)\n    f=factorial(k)\n    return sum(f // reduce(mul,[ factorial(v) for v in list(Counter(p).values()) if v > 1 ] + [1]) for p in tups)\n\ndef get_factor_tuples(n,k):\n    pair = {(n,)}\n    for kp in range(1,k):\n        k_tup = set(x for x in pair if len(x)==kp)\n        for grp in k_tup: \n            for i,e in enumerate(grp):\n                g = grp[:i]+grp[i+1:]\n                if e not in primefactors(n):\n                    for a in factors(e):\n                        if 1<a<e:\n                            b = e//a\n                            pair.add(tuple(sorted((a,b)+g)))\n    return pair\n\n@lru_cache(maxsize=None)\ndef primefactors(n):\n    i,f = 3,[]\n    while n&1 == 0:\n        f.append(2)\n        n = n >> 1\n    while i * i <= n:\n        if n % i: i+=2\n        else:\n            f.append(i)\n            n//=i\n    if n > 1: f.append(n)\n    return f\n\n@lru_cache(maxsize=None)\ndef factors(n):\n    pf = primefactors(n)\n    return { 1,n } | { reduce(mul,x) for z in range(1,len(pf)) for x in combinations(pf,z) }\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'multiply'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 8078