28003: A Lesson Learned

One of the limitations of the free version of the Airtable online database is that it only allows about 1000 records per database. Once the limit is reached, you need to create a new database to accommodate your needs. I've been using Airtable for years now and I am into my third database:

• Diurnal Age Part 1
• 1198 records (over the limit)
• started on March 17th 2017
• closed on June 26th 2021
  
  
• Diurnal Age Part 2
• 923 records
• started on June 27th 2021
• closed on January 2nd 2024
  
  
• Diurnal Age Part 3
• 704 records currently
• started on January 3rd 2024
  
  

This is an impressive combined database containing about 3000 records. However, the records are in three disconnected databases and once I move on to a new database, I've tended to ignore the earlier databases. However, I realised the folly of doing this when confronted with something interesting to say about the number associated with my diurnal age today: \( \textbf{28003}\).

It's not dark yet but it's getting there

The OEIS, Mathematical Meanderings (my Mathematics blog), Bespoken for Sequences and Diurnal Age Part 3 (in short all my usual sources) had nothing useful to contribute. It then occurred to me to look back at my Diurnal Age Part 2 database and I was pleasantly surprised at what I found therein. Let's begin:

• \( \textbf{28003} \) is a member of OEIS A179248: numbers that have 8 terms in their Zeckendorf representation: [3, 8, 89, 233, 610, 2584, 6765, 17711]. There is in face a cluster of sequence members nearby, namely 27800, 27802, 27804, 27807, 27808, 27809 and 27811.
  
  
• \( \textbf{28003}\) is a member of OEIS A157344: semiprimes that are the product of two distinct Sophie Germain primes. Approximately 2.52% of numbers in the range up to 40,000 satisfy this criterion. Here the primes are 41 and 683 and we have:
• 41 x 2 + 1 = 83 which is prime
• 683 x 2 + 1 = 1367 which is prime
  
  
• \( \textbf{28003}\) is a member of OEIS A082957: numbers \(n\) such that \( \sigma(2n) < \sigma(2n+1) \). These numbers total 5.52% in the range up to 40,000. There are 2301 primes and 3169 composites among the 5470 first terms. Here we have:
• \( \sigma(2 \times 28003) = \sigma(56006) = 86184\)
• \( \sigma(2 \times 28003 + 1) = \sigma(56007) = 94848\)
  
  
• \( \textbf{28003} \) is a member of OEIS A134252: positions of 2 after decimal point in decimal expansion of \( \frac{1}{\pi}\). 

Diurnal Age Part 1 yielded no results but as can be seen Diurnal Age Part 2 revealed membership of \( \textbf{28003}\) in four different OEIS sequences.