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An esoteric algorithm is a strange, confusing, or humorous algorithm that deviates from traditional algorithmic concepts in some way. An esoteric algorithm may be overly complicated, have unneeded results, or be just plain bonkers. An esoteric algorithm may be dependent on (accepting, returning, or using intermediately) an esoteric data structure.
The definition of an Esoteric algorithm is very broad, which allows for endless algorithms fulfilling it. Take the Hazard Algorithm, proposed by User: Areallycoolusername as an example. It's optimized for use with the Esoteric data structure he proposed, called the "Arch". It's an algorithm for calculating the first few digits in the "Hazard Integer Sequence" (also devised by Areallycoolusername), all the prime numbers in the Unlucky Number Sequence, multiplied by 3. The algorithm starts by storing a user-defined amount of numbers in the Unlucky Sequence in the Arch. Then, the primes in the set are separated, multiplied by 3, and then put in the Arch, replacing the original Unlucky numbers in the process.
Esoteric algorithms can be used to create new esoteric programming languages in general. For example, you can create an algorithm that gives you random keywords or characters for a new esoteric programming language depending on the contents of a string of integers given as input. As said before, the possibilities are endless, and therefore, it can help break the loop of monotonous languages, such as the brainfuck derivatives.