What are Semi R-Primes?

Prime numbers, as you may know, are integers greater than one that have no divisors other than the integer itself and one. For example, 83 is a prime since the only two integers that divide (leave a remainder of zero) 83 are 1 and 83.

A semi-prime is an integer that is a product of exactly two primes, not necessarily distinct. For example, 15 is a semi-prime since its prime factorization is \(3 \times 5\) (2 prime numbers). However, 20 is not a semi-prime because its prime factorization is \(2 \times 2 \times 5\) (3 prime numbers even though there are two distinct primes).

Now, consider this sequence: \(4, 6, 9, 22, 33, 55, 77, 111, 121, 141, …\) Do you notice that all of these numbers are palindromic? That is, they can be read as the same number forwards and backwards? But can you also prove to yourself that this sequence contains palindromic numbers that are semi primes? Take for example, 111. This number is a semi-prime because \(111 = 3\times 37\). Cool, huh? This sequence is published in the OEIS with designation A046328.

Now, I find it a shame that such a beautiful sequence doesn’t have a name. Thus, I propose that palindromic semi-primes be called “semi r-primes”. The “r” stands for “reversible” and the name “semi r-primes” is a palindrome!

Let’s make this happen!

Update 2/19/2019

The singular form should be called an “emi r-prime” and the plural is “semi r-primes”. If we want, we can call the singular an “emir prime” to give it an air of regalness.

Let me also say that the physicists have “mho” as the unit for \(\frac{1}{\mbox{ohm}}\) where an ohm is the unit for resistance. So by gum, it’s a moral imperative that we mathematicians have emir prime and semi r-primes.


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