A Special Class of Semiprimes

A well-known factorisation involves the sum of two cubes:$$ a^3 + b^3=(a+b)(a^2-ab+b^2)$$Assuming that both \(a\) and \(b\) are positive integers, then this sum of two cubes is a semiprime if \(a+b\) and \(a^2-ab+b^2\) are both prime. It's interesting to explore when this happens.

Let's begin with \(b=1\) and restrict ourselves to semiprimes between 4 and 4000. There are only seven that qualify and they are:$$9, 65, 217, 4097, 5833, 10649, 21953 $$These numbers form the initial terms of OEIS A237040. Their factorisations are as follows:$$ \begin{align} 9 &= 3^2 = 2^3 + 1^3\\65 &= 5 \times 13 = 4^3 + 1^3\\217 &= 7 \times 31 = 6^3 + 1^3\\4097 &= 17 \times 241 = 16^3 + 1^3\\5833 &= 19 \times 307 = 18^3 + 1^3\\10649 &= 23 \times 463 = 22^3 + 1^3\\21953 &= 29 \times 757 = 28^3 + 1^3 \end{align}$$The next is \(b=8\) which yields six numbers:$$35, 133, 737, 1339, 3383, 24397$$These numbers factorise as follows:$$ \begin{align} 35 &= 5 \times 7 = 3^3 + 2^3\\133 &= 7 \times 19 = 5^3 + 2^3\\737 &= 11 \times 67 = 9^3 + 2^3\\1339 &= 13 \times 103 = 11 ^3 + 2^3\\3383 &= 17 \times 199 = 15^3 + 2^3\\24397 &= 31 \times 787 = 29^3 + 2^3 \end{align} $$These terms are not listed in the OEIS database. Next we'll let \(b=27\). This yields eight numbers and we see that 35 makes a reappearance as the 2 and 3 swap places. The numbers are:$$35, 91, 1027, 2771, 8027, 17603, 21979, 39331 $$These numbers are not listed in the OEIS database and their factorisations are:$$ \begin{align} 35 &= 5 \times 7 = 2^3 + 3^3\\91 &= 7 \times 13 = 4^3 + 3^3\\1027 &= 13 \times 79 = 10^3 + 3^3\\2771 &= 17 \times 163 = 14^3 + 3^3\\8027 &= 23 \times 349 = 20^3 + 3^3\\17603 &= 29 \times 607 = 26^3 + 3^3\\21979 &= 31 \times 709 = 28^3 + 3^3\\39331 &= 37 \times 1063 = 34^3 + 3^3 \end{align} $$Here is a permalink to an algorithm that allows further exploration using additional values of \(b\). A further avenue for investigation would be to consider semiprimes that are a difference of two cubes since we know that:$$ a^3 - b^3=(a-b)(a^2+ab+b^2)$$Using \(b=-1\), we find that the following semiprimes qualify:$$26, 215, 511, 1727, 2743, 7999, 13823$$These numbers form the initial members of OEIS A242262. Their factorisations are as follows:$$ \begin{align} 26 &= 2 \times 13 = 3^3 -1^3\\215 &= 5 \times 43 = 6^3 - 1^3\\511 &= 7 \times 73 = 8^3 - 1^3\\1727 &= 11 \times 157 = 12^3 - 1^3\\2743 &= 13 \times 211 = 14^3 - 1^3\\7999 &= 19 \times 421 = 20^3-1^3\\13823 &= 23 \times 601 = 24^3-1^3 \end{align} $$