Revisiting Fibodiv and Repfigit Numbers

On the 21st January 2024, I made a post titled Fibodiv Numbers and today I'm revisiting this topic because $\mathbf{\text{27801}}$, the number associated with my diurnal age today, is one such number. There aren't many of them. In the range up to one million, there are only 233. Up to 40,000 these are (permalink):

14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995, 8199, 8845, 9107, 9161, 9267, 9744, 10408, 11709, 12356, 12992, 13010, 14311, 14999, 15445, 15612, 16913, 17690, 18214, 18322, 18534, 19515, 19999, 20816, 20987, 21623, 22117, 23418, 24712, 24719, 26020, 27321, 27483, 27801, 28622, 29107, 29923, 29998, 30890, 31224, 32498, 32525, 33826, 33979, 35127, 36428, 36644, 37729, 39030

These numbers form OEIS A130792: numbers $k$ whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing $k$ itself. 

In that OEIS entry Mathematica code has been entered which is incomprehensible to me but fortunately Google Gemini (or similar) can be used to convert this code to Python that can then be run in SageMathCell. The Python code runs perfectly and quickly generates the 233 Fibodiv numbers. Gemini also provides an explanation of how the code works and additional help can be obtained if needed (link). This ease of converting from any programming language to a language of ones own choice is very useful.

So let's see how 27801 earns it right of inclusion into OEIS A130792 (permalink):$$27,801,828,1629,2457,4086,6543,10629,17172,27801$$We see how the two parts of 27801, 27 and 801, serve as seeds for a Fibonacci-like sequence that eventually generates the number 27801.

Now Repfigit numbers are similar to Fibodiv numbers and in fact the two digit Repfigit numbers are also Fibodiv numbers. I discussed these in my post On Turning 75 on April 3rd 2024. They belong to OEIS A007629: Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers): numbers $n$ with $k$ digits such that a Fibonacci-like sequence can be defined using as seeds the digits of $n$ and then at each step adding the last $k$ terms. If $n$ itself appears in the sequence, then it is a repfigit number. Up to 40000, the members of this sequence are (permalink):

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348

An example is 197 with three digits such that:$$1,9,7,17,33,57,107,197$$We can see that:$$\begin{array}{rl}1+9+7& =17\\ 9+7+17& =33\\ 7+17+33& =57\\ 17+33+57& =107\\ 33+57+107& =197\end{array}$$A full list of Keith numbers can be found at this site.