A Semiprime Rara Avis

A rara avis or rare bird is someone or something that is rare and this type of semiprime is indeed rare. It came to my attention recently when I looked at a number associated with my diurnal age that, at the time, was 27331 days old. This number is a semiprime and factorises to:$$27331= \underbrace{151}_{ \text{19th 4k+3 prime}} \times \underbrace{181}_{ \text{19th 4k+1 prime}}$$Numbers like 27331 form OEIS A048630 (permalink) and remember that \(4k-1 \equiv 4k+3\):

A048630

\(n\)-th 4\(k\)+1 prime times \(n\)-th 4\(k\)-1 prime.

The first 22 of these semiprimes are as follows:

• 1 --> 15 = 3 x 5

• 2 --> 91 = 7 x 13

• 3 --> 187 = 11 x 17

• 4 --> 551 = 19 x 29

• 5 --> 851 = 23 x 37

• 6 --> 1271 = 31 x 41

• 7 --> 2279 = 43 x 53

• 8 --> 2867 = 47 x 61

• 9 --> 4307 = 59 x 73

• 10 --> 5963 = 67 x 89

• 11 --> 6887 = 71 x 97

• 12 --> 7979 = 79 x 101

• 13 --> 9047 = 83 x 109

• 14 --> 11639 = 103 x 113

• 15 --> 14659 = 107 x 137

• 16 --> 18923 = 127 x 149

• 17 --> 20567 = 131 x 157

• 18 --> 24047 = 139 x 173

• 19 --> 27331 = 151 x 181

• 20 --> 31459 = 163 x 193

• 21 --> 32899 = 167 x 197

• 22 --> 40991 = 179 x 229

As can be seen, my next "experience" of such a semiprime will come when I am 31459 days old. This will occur on Monday, May 21st 2035, not long after my 86th birthday if I make it that far. So in terms of life experiences, marked by the passing of the days, most individuals will be lucky if they see 21 of them. In my case the 21st falls on Saturday, April 30th 2039, three weeks after my 90th birthday. Nobody will experience the 22nd such semiprime!

Working with the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime, we could generate other sequences such as for example the average of the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime. This sequence is not listed in the OEIS but it begins with 4, 10, 14, 24, 30, 36, 48, 54, 66, 78, 84, 90, 96, 108, 122, 138, 144, 156, 166, 178, 182, 204 (permalink). These sorts of sequences could be termed "hybrid" because they involve the combination of two related but separate sequences.

We could take the sum of the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime and add 1 to generate a sequence of odd numbers: 9, 21, 29, 49, 61, 73, 97, 109, 133, 157, 169, 181, 193, 217, 245, 277, 289, 313, 333, 357, 365, 409 etc. (permalink). Again, this is not listed in the OEIS. All sorts of combinations are possible between these and other sequences, the results of which might be of mathematical interest but most fall into the realm of recreational mathematics.

In a recent post, Yet Another Type of Prime on January 19th 2024, I examined hybrid sequences arising from the sum of the \(n\)-th prime and the \(n\)-th composite number as well as the sum of the  \(n\)-th prime and the \(n\)-th non-prime number. I went on to look at subsequences involving the primes arising within these summed sequences.