# Is always two times an even semiprime at a distance $1$ or prime to the closest previous odd semiprime?

This is an observation regarding the semiprimes, also named 2-almost primes, biprimes, or the product of two primes. This week I do not have a computer, only a tablet (hospitalized with a lot of free time to enjoy Maths!) so I just can read and use a pen and a notebook for calculations.

Basically it seems that two times an even semiprime ($2p$), this is $4p, p \in \Bbb P$ is located a a prime distance or distance $1$ of the previous odd semiprime of the list (except for the first element of the list, $4$, that has no odd semiprimes below $8$).

I was able to verify this for the first elements available on the OEIS list (linked above in the first paragraph) and then I took random samples from the link at OEIS to the big list of the first $10000$ elements.

E.g. The biprime $6 \cdot 2 = 12$ and the closest previous odd semiprime is $9$ at a distance $3$.

For instance, a similar test like "the double of a non even semiprime is at a distance $1$ or prime of the closest previous odd semiprime" does not hold. There are quick counterexamples.

I would like to share the following questions:

1. Is there a counterexample? I can not run a computer test, if somebody could confirm if the observation is true or false in some extent that would be great.

2. What would be behind this kind of property (quite similar to the Fortunate numbers conjecture). Thank you!

I wrote a (messy) Mathematica program, which told me the smallest counter example is $218$. The largest odd semiprime smaller than $218\times2$ is $427$, and $218\times2-427=9$ which is not prime.
A couple other counter examples are $466$ and $746$.
• Agree with your findings. The odd primes $p$ such that the least $j>0$ that makes $4p-j$ an odd semiprime is composite, are 109, 233, 373, 523, 601, 683, 751, 863, 883, 1231, 1237, 1361, 1381, 1399, 1423, 1619, 1699, 1747, 1777, 1873, 2011, .... In the subsequence where $p$ equals 373, 523, 1423, 2953, 3067, 3137, 3539, 3583, 4049, 4513, 4877, 5113, 5381, 6529, 7057, 8831, 8999, 9041, 9791, 9811, ... the $j$ value is not a prime power either. – Jeppe Stig Nielsen Mar 23 at 11:44