### Background

I was reading a [book](http://www.amazon.co.uk/Things-Make-Do-Fourth-Dimension/dp/1846147646/) recently, "Things to Make and Do in the Fourth Dimension" by comedian and mathematician Matt Parker, and in the first chapter of the book Matt talks about problems he likes to solve in his head to take his mind off the fact that he is in his dentist's chair, we've all been there!

The problem he talks about relates to polydivisible numbers, and I thought a kata should be written on the subject as it's quite interesting. (Well it's interesting to me, so there!)

### Polydivisib... huh what?

So what are they?

A polydivisible number is divisible in an unusual way. The first digit is cleanly divisible by `1`, the first two digits are cleanly divisible by `2`, the first three by `3` and so on.

The interesting thing about polydivisiblity is that it relates to the underlying number, but not the base it is written in, so if aliens came to Earth and used base `23` (`11` fingers on one hand and `12` on the other), no matter what squiggles they use to write numbers, they would find the same numbers polydivisible!

### Polydivisibilty Example:

Let's do a worked example to clear up any questions ...

Starting wih the number `1,232` in base `10` then:

```
 1232
 1    /1 = 1    Yay!
 12   /2 = 6    Yay!
 123  /3 = 41   Yay!
 1232 /4 = 308  Yay!
```

Thus `1,232` is a polydivisible number in base `4` and above.

However starting wih the number `123,220` and using base `10` then:

```
 123220
 1      /1 = 1            Yay!
 12     /2 = 6            Yay!
 123    /3 = 41           Yay!
 1232   /4 = 308          Yay!
 12322  /5 = 2464.4       Oh no, that's not a round number!
 123220 /6 = 220536.333r  Oh no, that's not a round number!
```

Thus `123,220` is not a polydivisible base 10 number, but what about in another base?

Again starting wih the number `123,220` and using base `6` then:

```
 base 6   base 10
 1      = 1       -> 1     /1 = 1     Yay!
 12     = 8       -> 8     /2 = 4     Yay!
 123    = 51      -> 51    /3 = 17    Yay!
 1232   = 308     -> 308   /4 = 77    Yay!
 12322  = 1850    -> 1850  /5 = 370   Yay!
 123220 = 11100   -> 11100 /6 = 1850  Yay!
```

Thus `123,220` is a polydivisible base `6` number (and a polydivisible base `10` number when converted to `11100` in base `10`).

### Kata

In this kata you must implement two methods: `is_polydivisible(n, b)` and `get_polydivisible(n, b)`.

The first `is_polydivisible(n, b)` will return `True` if `n` is polydivisible in base `b` or `False` if not.

The second `get_polydivisible(n, b)` will return the `n`th polydivisible number using base `b`, the first polydivisible number is of course always `0`.

You can assume that all inputs are valid.

```if:haskell
All necessary arithmetic can be done in `Int` range.
```

### Kata Examples:

```python
    is_polydivisible("1232", 10)   # => True
    is_polydivisible("123220", 10) # => False
    is_polydivisible("123220", 6)  # => True
    get_polydivisible(22, 10)      # => "32"
    get_polydivisible(22, 16)      # => "1A"
    get_polydivisible(42, 16)      # => "42"
```

#### A Note on Bases

The maximum base used is base `62`, and uses characters in the following order `[0-9][A-Z][a-z]` to denote its digits, base `n` will use the first `n` characters of this sequence.
```if-not:haskell
A constant CHARS has been declared with this sequence for you.
```

```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 = 'CHARS="0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"\n\ndef to_dec(s,b):\n    n = 0\n    for i,v in enumerate(s[::-1]):\n        n += CHARS.index(v)*b**i\n    return n\n\ndef to_base(n,b):\n    r = \'\'\n    while n >= b:\n        n,x = divmod(n,b)\n        r += CHARS[:b][x]\n    r += CHARS[:b][n]\n    return r[::-1]\n         \ndef is_polydivisible(s,b):\n    return all(not to_dec(s[:i],b)%i for i in range(1,len(s)+1))\n\ndef get_polydivisible(x,b):\n    c = 0\n    n = 0\n    while c<x:\n        try:\n            if is_polydivisible(to_base(n,b),b):\n                c += 1\n        except:\n            pass\n        n += 1\n    return to_base(n-1,b) or \'0\''
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'is_polydivisible'
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: 6368