Problem 58019. Factor a number into Fermi-Dirac primes

Created by ChrisR
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Cody Problem 58018 asked you to list the Fermi-Dirac primes, which are prime powers with exponents that are powers of 2. As noted there, the name comes from an analogy with particle physics because every number can be written as the product of a unique subset of the Fermi-Dirac primes, in which the Fermi-Dirac primes appear at most once. For example, . 
Write a function analogous to factor(n) that factors the number n into Fermi-Dirac primes. List the Fermi-Dirac primes in ascending order. Take the factorization of 1 to be 1. 

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Last Solution submitted on Mar 02, 2025

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Dyuman Joshi on 8 Dec 2024

What should be the output for numbers such as 256?
[4 64]? [16 16]?

Or 1024?
[4 256]? [4 4 64]? [4 4 4 16]? [16 64]? [4 16 16]?

Does it have to be a particular pairing(s)? If yes, how so?
Or will any/all of them work?

Christian Schröder on 8 Dec 2024

@Dyuman Here's my two cents:

256 = [256]. [4 64] is not valid since 64 is not a prime power where the exponent is a power of two. [16 16] is out since - I assume - you're supposed to simplify as much as possible. Same for [2 2 2 2 2 2 2 2], [4 4 4 4] and so on. But you're right that this additional requirement is needed to ensure uniqueness.

1024 = [4 256] using the same reasoning.

ChrisR on 28 Dec 2024

I added the requirement that the FD primes appear in the factorization at most once. This requirement resembles a difference between the Fermi-Dirac and the Bose-Einstein distributions of particles.