A Range Limit of 40000: Why?

In many of my posts when I'm considering sequences, I only look at sequence members whose values do not exceed 40000. Why? The answer to this question relates to the reason that I set this blog up in the first place. It was designed primarily to post about interesting sequences associated with the numbers marking my diurnal age.

If we divide 40000 by 365.2425 (the average number of days in a year) we get slightly more than 109.5 years and there are not many people who live to that ripe old age. Here are milestones, one might say, on the road to oblivion.

Quite a few people won't see 30000 days let alone 40000 but many will and hence the realistic upper limit to the numbers that I normally investigate. The focus of my blog posts is generally the number associated with my diurnal age and the sequences that it can be connected to. For example, today I am \( \textbf{27920} \) days old.

This number found its way into a sequence that I created that involves gapful numbers with the property that not only does the number formed by the concatenation of the first and last digit divide the number but this concatenated number is also the sum of the number's digits. Thus we have:$$ \begin{align} \frac{27920}{20} &=1396 \\ \\ 2 + 7 + 9 + 2 + 0 &= 20 \end{align}$$I described this sequence in a post titled Gapgul Numbers in December of 2024. Interestingly, 27920 also has the property that it has 20 divisors. Only 19 numbers satisfy this additional criterion in the range up to 40000. The conditions to be met are:

• the number is gapful meaning the number formed by concantenating the first and last digits divides the number without remainder
• the sum of the number's digits equals the number formed by the concatenated first and last digits
• the number of divisors of the number equals its sum of digits (and the concatenated number)

Here are the numbers:

1548, 1812, 1908, 10188, 10548, 11268, 12252, 12612, 12708, 13428, 14052, 14412, 15138, 18108, 21984, 26480, 27920, 29360, 39996

Here are the details:

  number        factor          concat   dividend   S0D   divisors

  1548     2^2 * 3^2 * 43         18       86         18    18
  1812     2^2 * 3 * 151          12       151        12    12
  1908     2^2 * 3^2 * 53         18       106        18    18
  10188    2^2 * 3^2 * 283        18       566        18    18
  10548    2^2 * 3^2 * 293        18       586        18    18
  11268    2^2 * 3^2 * 313        18       626        18    18
  12252    2^2 * 3 * 1021         12       1021       12    12
  12612    2^2 * 3 * 1051         12       1051       12    12
  12708    2^2 * 3^2 * 353        18       706        18    18
  13428    2^2 * 3^2 * 373        18       746        18    18
  14052    2^2 * 3 * 1171         12       1171       12    12
  14412    2^2 * 3 * 1201         12       1201       12    12
  15138    2 * 3^2 * 29^2         18       841        18    18
  18108    2^2 * 3^2 * 503        18       1006       18    18
  21984    2^5 * 3 * 229          24       916        24    24
  26480    2^4 * 5 * 331          20       1324       20    20
  27920    2^4 * 5 * 349          20       1396       20    20
  29360    2^4 * 5 * 367          20       1468       20    20
  39996    2^2 * 3^2 * 11 * 101   36       1111       36    36