A well-known factorisation involves the sum of two cubes:Assuming that both and are positive integers, then this sum of two cubes is a semiprime if and are both prime. It's interesting to explore when this happens.
Let's begin with and restrict ourselves to semiprimes between 4 and 4000. There are only seven that qualify and they are:These numbers form the initial terms of OEIS A237040. Their factorisations are as follows:The next is which yields six numbers:These numbers factorise as follows:These terms are not listed in the OEIS database. Next we'll let . This yields eight numbers and we see that 35 makes a reappearance as the 2 and 3 swap places. The numbers are:These numbers are not listed in the OEIS database and their factorisations are:Here is a permalink to an algorithm that allows further exploration using additional values of . A further avenue for investigation would be to consider semiprimes that are a difference of two cubes since we know that:Using , we find that the following semiprimes qualify:These numbers form the initial members of OEIS A242262. Their factorisations are as follows:
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