Squarefree Semiprime Chains

The number associated with my diurnal age today, 26734, has an interesting property that qualifies it for membership in OEIS A177215:

A177215 Numbers \(k\) that are the products of two distinct primes such that \(2k-1, 4k-3, 8k-7, 16k-15, 32k-31 \text{ and } 64k-63\) are also products of two distinct primes.

Let's confirm that:

There are 281 members of OEIS  A177215 up to one million. I'll use the highest of them, 998185, as an example. It has the properties shown below:

What if we raise the stakes so to speak and apply the additional criterion that \(128k-123\) is also a semiprime with two distinct prime factors (in other words it's squarefree)? Well, the number shrinks to 81 in the range up to one million. The highest number that qualifies in that range is 991237 with the following properties:

Let's push thing further. How many numbers qualify once we add the additional criterion of \(256k-255\) being a squarefree semiprime? The number now reduces to 29, the highest of which remains our 991237. The revised table is shown below:

Let's keep pushing. How many numbers quality if we add the additional criteria that \(512k-511\) must be a squarefree semiprime? The number is now down to 14 and 991237 is still at the top. Here's the revised table.

Pushing further to include \(1024k-1023\), we find that only four numbers qualify in the range up to one million. These are 173311, 346621, 464245 and 563326. The table below shows the properties for 563326:

Proceeding logically, we now look at the criterion \(2056k-2055\) and we find that there is only one man left standing and that is 173311. Its table properties is shown below:

So overall, an interesting journey that ends with 173311 because this number doesn't pass the \(4096k-4095\) test. Of course, the journey never ends. We must ask what is the smallest number greater than one million that satisfies the criterion that \(4096k-4095\) is a squarefree semiprime? Well that number is 2212801. It's properties are shown below:

I'll stop there but here is a SageMathCell permalink to the program that I was using to discover these numbers. Feel free to experiment further.

ADDENDUM:

After making this post, I discovered that I'd covered much of the same territory in a post from October 17th 2018 titled Semiprime Chains. It will teach me to look back over my previous posts before posting. I've made so many posts of the years that it's easy to forget my earlier posts. One thing that is different between the two posts is the code that I used. The latter code is far more succinct. It's just as well that I looked back because the embedded code that I'd used was outdated. It contained the superseded print \(n\) command instead of the Python 3 print(\(n\)) command. I've corrected that now.

Here are the links to my previous posts on semiprimes:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• Semiprime Factor Ratios
• More about Golden Semiprimes
• 2019: A Numerical Profile
• Semiprime Chains
• Another Look at Semiprimes
• Visualisation of Semiprimes
• Semiprimes that Approximate Whole Numbers

I'm surprised at how many posts I've made over the years about semiprimes. It would be interesting to make a small booklet on the topic given the amount of material that I've accumulated.