I am currently working on a sequence, $a_n$, that is defined as follows:
$$a_n\text{ is the smallest product of }n\text{ distinct odd primes, }m=p_1p_2\dots p_n\text{, such that }\frac{d+\frac{m}{d}}{2}\text{ is prime for all }d\text{ dividing }m.$$
This sequence is currently known to five terms: $3, 21, 105, 1365, 884037$ (OEIS A128281). I have coded a program in an attempt to find a sixth term. During my search, I was able to determine that all members of the sequence except the first are congruent to $1{\pmod{4}}$. My best method currently is to test every number congruent to $1{\pmod{4}}$ starting at $255257 = 3\cdot5\cdot7\cdot11\cdot13\cdot17+2$. My program tests first if each number is a product of six distinct odd primes, then tests if it fits the divisor requirements for the sequence. With my program I have determined that $a_6>10^9$. Are there any optimizations I could make so that my program takes less time to search? Is there any reason to suspect that $a_6$ does not exist? If so, would this be the case for further terms as well?