Return the number of permutations of 1 to n so that prime numbers are at prime indices (1-indexed.)
(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)
Since the answer may be large, return the answer modulo 10^9 + 7.
 
Example 1:
Input: n = 5
Output: 12
Explanation: For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.

Example 2:
Input: n = 100
Output: 682289015

 
Constraints:

1 <= n <= 100

```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 = "class Solution:\n    def numPrimeArrangements(self, n: int) -> int:\n        p_nums = []\n        c_nums = []\n        dp = [True for _ in range(n+1)]\n        dp[0] = False\n        dp[1] = False\n        for i in range(2, n+1):\n            if dp[i] == True:\n                cnt = 2\n                while i*cnt < n+1:\n                    dp[i*cnt] = False\n                    cnt += 1\n        \n        for i in range(1, n+1):\n            if dp[i]:\n                p_nums += [i]\n            else:\n                c_nums += [i]\n\n        #return (self.fact(len(p_nums))*self.fact(len(c_nums)))%(10**9+7)  ===ok\n        #模运算规则：(a * b) % m = ((a % m) * b) % m 证明方法请自行搜索\n        return (self.fact(len(p_nums) % (10**9+7))*self.fact(len(c_nums)) % (10**9+7) )\n    \n    def fact(self, n):\n        if n <= 1:\n            return 1\n        else:\n            return n*self.fact(n-1)\n\n        \n    '''\n    题目描述\n\n中文：质数排列\n英语：Prime Arrangements\n\n解题思路\n\n理解题意，是要求1到n的排列组合数，但并不是全排列，而是1到n中所有质数个数的阶乘和非质数个数的阶乘的乘积并对10^9 + 7即1000000007取模运算的结果。\n以n=5为例，1到5中有2、3、5共3个质数，非质数则有1、4共2个，则结果为(3! * 2!) % 1000000007 = 12\n\n代码如下\n\n需要理解的点 模运算规则：(a * b) % m = ((a % m) * b) % m 证明方法请自行搜索\n\n将数字分为质数和非质数两部分。\n假设有 p个质数，答案就是 p!⋅(n−p)!\n\n     '''\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numPrimeArrangements'
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: 6170