If n term is even, the next term is floor value of square root of n .
    If n is odd, the next term is floor value of 3 time the square root of n.

https://en.wikipedia.org/wiki/Juggler_sequence
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)
import math


def juggler_sequence(number: int) -> list[int]:
    """
    >>> juggler_sequence(0)
    Traceback (most recent call last):
        ...

in a pattern of dots consisting of the outlines of regular
hexagons with sides up to n dots, when the hexagons are
overlaid so that they share one vertex.

https://en.wikipedia.org/wiki/Hexagonal_number
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)


def hexagonal(number: int) -> int:
    """
    :param number: nth hexagonal number to calculate
    :return: the nth hexagonal number

and λ(n) is 1 if n is the product of an even number of prime numbers,
and -1 if it is the product of an odd number of primes.

https://en.wikipedia.org/wiki/Liouville_function
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)
from maths.prime_factors import prime_factors


def liouville_lambda(number: int) -> int:
    """
    This functions takes an integer number as input.

both n and n+2 are prime.

Examples of Twin pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ...
https://en.wikipedia.org/wiki/Twin_prime
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)
from maths.prime_check import is_prime


def twin_prime(number: int) -> int:
    """
    # doctest: +NORMALIZE_WHITESPACE
    This functions takes an integer number as input.

there exists an integer m such that n = m * (m + 1)

Examples of Proic Numbers: 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110 ...
https://en.wikipedia.org/wiki/Pronic_number
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)


def is_pronic(number: int) -> bool:
    """
    # doctest: +NORMALIZE_WHITESPACE
    This functions takes an integer number as input.
    returns True if the number is pronic.

the square of n "ends" in the same digits as n itself.

Examples of Automorphic Numbers: 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, ...
https://en.wikipedia.org/wiki/Automorphic_number
"""

# Author : Akshay Dubey (https://github.com/itsAkshayDubey)
# Time Complexity : O(log10n)


def is_automorphic_number(number: int) -> bool:
    """
    # doctest: +NORMALIZE_WHITESPACE
    This functions takes an integer number as input.

- [docs] Fix the link for the sx props page (#27202) @mnajdova
- [docs] Fix theme context example code (#27053) @moshfeu
- [docs] Fix typo in CONTRIBUTING.md (#27218) Ayush Dubey
- [docs] Fix typos (#27074) @michaldudak
- [docs] Improve nav semantics (#27138) @eps1lon
- [docs] Migrate Ad\* components to emotion (#27159) @mnajdova
- [docs] Migrate rest of the docs to emotion (#27184) @mnajdova

PragmaTwice, pshiko, Qwerty71, R Gomathi, Rahul Huilgol, Richard Xiao, Rick
Wierenga, Roberto Rosmaninho, ruchit2801, Rushabh Vasani, Sami, Sana Damani,
Sarvesh Dubey, Sasan Jafarnejad, Sergii Khomenko, Shane Smiskol, Shaochen Shi,
sharkdtu, Shawn Presser, ShengYang1, Shreyash Patodia, Shyam Sundar Dhanabalan,
Siju Samuel, Somyajit Chakraborty Sam, Srihari Humbarwadi,