AD and BC Numbers

In my quest to find interesting properties for the successive numbers that constitute my diurnal age, I usually make use of the OEIS and Numbers Aplenty. Occasionally these resources throw up little of interest and, after exhausting additional resources, I'm left to come up with some creative angle of my own.

Today's number, 26541, presented such a challenge and I'm proud to say I rose to the occasion. While professional mathematicians will sneer, recreational mathematical enthusiasts may appreciate my resourcefulness. Here is what I came up with. Decimal numbers when converted to hexadecimal can end in AD and the thought struck me that such numbers would form a sequence. I've entered this sequence into my own private database of sequences. Here it is:

S037: Anno Domini (AD) numbers: decimal numbers that when converted to hexadecimal contain at least three digits satisfying the following criteria: 

• the second last digit is A
• the last digit is D
• all remaining digits are between 0 and 9

Up to 40000, the list of such numbers is as follows:

429, 685, 941, 1197, 1453, 1709, 1965, 2221, 2477, 4269, 4525, 4781, 5037, 5293, 5549, 5805, 6061, 6317, 6573, 8365, 8621, 8877, 9133, 9389, 9645, 9901, 10157, 10413, 10669, 12461, 12717, 12973, 13229, 13485, 13741, 13997, 14253, 14509, 14765, 16557, 16813, 17069, 17325, 17581, 17837, 18093, 18349, 18605, 18861, 20653, 20909, 21165, 21421, 21677, 21933, 22189, 22445, 22701, 22957, 24749, 25005, 25261, 25517, 25773, 26029, 26285, 26541, 26797, 27053, 28845, 29101, 29357, 29613, 29869, 30125, 30381, 30637, 30893, 31149, 32941, 33197, 33453, 33709, 33965, 34221, 34477, 34733, 34989, 35245, 37037, 37293, 37549, 37805, 38061, 38317, 38573, 38829, 39085, 39341

All the numbers differ by 256 except for a regular jump of 1792 or 7 x 256 after every 9th number. In fact, all the numbers equal 173 mod 256 and 173 is the decimal value of AD. The first number in the sequence, 429 = 256 + 173, converts to 1AD while the last number, 39341 = 9 x 4096 + 6 x 256 + 173, converts to 99AD. The number associated with my diurnal age, 26541, converts to 67AD. Here is a permalink to the code that I used in SageMathCell.

The logical extension of course is to identify BC numbers and that is what I've done in another private sequence:

S038: Before Christ (BC) numbers: decimal numbers that when converted to hexadecimal contain at least three digits satisfying the following criteria: 

• the second last digit is B
• the last digit is C
• all remaining digits are between 0 and 9

Up to 40000, the members of this sequence are:

444, 700, 956, 1212, 1468, 1724, 1980, 2236, 2492, 4284, 4540, 4796, 5052, 5308, 5564, 5820, 6076, 6332, 6588, 8380, 8636, 8892, 9148, 9404, 9660, 9916, 10172, 10428, 10684, 12476, 12732, 12988, 13244, 13500, 13756, 14012, 14268, 14524, 14780, 16572, 16828, 17084, 17340, 17596, 17852, 18108, 18364, 18620, 18876, 20668, 20924, 21180, 21436, 21692, 21948, 22204, 22460, 22716, 22972, 24764, 25020, 25276, 25532, 25788, 26044, 26300, 26556, 26812, 27068, 28860, 29116, 29372, 29628, 29884, 30140, 30396, 30652, 30908, 31164, 32956, 33212, 33468, 33724, 33980, 34236, 34492, 34748, 35004, 35260, 37052, 37308, 37564, 37820, 38076, 38332, 38588, 38844, 39100, 39356

Again, all the numbers differ by 256 except for a regular jump of 1792 or 7 x 256 after every 9th number. Here, all the numbers equal 188 mod 256 and 188 is the decimal value of BC. The first number in the sequence, 444 = 256 + 188, converts to 1BC while the last number, 39356 = 9 x 4096 + 9 x 256 + 188, converts to 99BC. Here is a permalink to the code that I used in SageMathCell. 

I know that BCE and CE can be used instead of BC and AD but I refuse to use this system. Anyone so inclined however, could modify my existing algorithms or create algorithms of their own in order to generate hexadecimal numbers ending in BCE and CE.

The numbers don't have to be necessarily hexadecimal because the letters used in my system are A, B, C and D and thus will work for bases 14 and 15 and 17 up to 36. I just chose hexadecimal because of its widespread use in technology. 

It's interesting that 67 AD has associations with Christianity because this year marked the end of St. Paul's journeying around the Mediterranean. See Figure 1.

Figure 1: source

This post marks my 100th post for the year 2021, an annual record that I've never even approached before. Previously, my annual totals were:

• 2020: 68 posts
• 2019: 56 posts
• 2018: 68 posts
• 2017: 36 posts
• 2016: 42 posts
• 2015: 12 posts

I started this blog in September of 2015, shortly after my retirement from teaching.