## 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 = 'from math import gcd, factorial\nfrom random import randint\n\ndef primesfrom2to(n):\n    # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n/3035188#3035188\n    """\n    Returns an array of primes, 2 <= p < n.\n    """\n    sieve = [True]*(n//2)\n    for i in range(3, int(n**0.5)+1, 2):\n        if sieve[i//2]:\n            sieve[i*i//2::i] = [False]*((n- i*i -1)//(2*i) + 1)\n    return [2] + [2*i + 1 for i in range(1, n//2) if sieve[i]]\n\n\nbound = 7100\nsmall_primes = primesfrom2to(bound)\nsmall_primes_set = set(small_primes)\n\ndef factorint(n):\n    """\n    Returns a dict containing the prime factors of n as keys and their\n    respective multiplicities as values.\n    """\n    factors = dict()\n\n    limit = int(n**0.5)+1\n    for prime in small_primes:\n        if prime > limit:\n            break\n        while n%prime == 0:\n            if prime in factors:\n                factors[prime] += 1\n            else:\n                factors[prime] = 1\n            n //= prime\n\n    if n >= 2:\n        large_factors(n, factors)\n    return factors\n\n\ndef large_factors(n, factors):\n    while n > 1:\n        if isprime(n):\n            if n in factors:\n                factors[n] += 1\n            else:\n                factors[n] = 1\n            break\n        \n        factor = pollard_brent(n)\n        large_factors(factor, factors)\n        n //= factor\n\n\ndef pollard_brent(n):\n    # https://comeoncodeon.wordpress.com/2010/09/18/pollard-rho-brent-integer-factorization/\n    """\n    Returns a factor of n. The returned factor may be a composite number.\n    """\n    if n%2 == 0:\n        return 2\n\n    y, c, m = randint(1, n-1), randint(1, n-1), randint(1, n-1)\n    g, r, q = 1, 1, 1\n    while g == 1:\n        x = y\n        for i in range(r):\n            y = (pow(y, 2, n)+c)%n\n\n        k = 0\n        while k < r and g == 1:\n            ys = y\n            for i in range(min(m, r-k)):\n                y = (pow(y, 2, n)+c)%n\n                q = q*abs(x-y)%n\n            g = gcd(q, n)\n            k += m\n        r *= 2\n\n    if g == n:\n        while True:\n            ys = (pow(ys, 2, n)+c)%n\n            g = gcd(abs(x-ys), n)\n            if g > 1:\n                break\n\n    return g  \n\n\ndef isprime(n):\n    # https://en.wikipedia.org/wiki/Miller–Rabin_primality_test#Deterministic_variants\n    if n%2 == 0:\n        return False\n    elif n < bound:\n        return n in small_primes_set\n\n    d, r = n-1, 0\n    while d%2 == 0:\n        d //= 2\n        r += 1\n\n    for a in (2, 3, 5, 7):\n        x = pow(a, d, n)\n        if x == 1 or x == n-1:\n            continue\n\n        for i in range(r-1):\n            x = pow(x, 2, n)\n            if x == n-1:\n                break\n        else:\n            return False\n\n    return True\n\n\ndef partitions(n, k):\n    """\n    Returns the number of ways to write n as the sum of k nonnegative \n    integers. Order matters.\n    """\n    result = 1\n    for i in range(n+1, n+k):\n        result *= i\n    return result//factorial(k-1)\n\n\ndef multiply(n, k):\n    total, factors = 1, factorint(n)\n    for exponent in list(factors.values()):\n        total *= partitions(exponent, k)\n    return total\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: 5860