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 \( \textbf{27801} \), 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:$$ \begin{align} 1 + 9 + 7 &= 17\\ 9+7+17 &= 33 \\ 7 + 17 + 33 &=57 \\ 17 + 33 + 57 &=107 \\33+57+107 &= 197 \end{align} $$A full list of Keith numbers can be found at this site. 

On Turning 75

On April 3rd 2024, I turned 75 years old. I like the graphic above that is meant to represent 75%. This translates nicely into years as well, because the maximum span of human life is more or less 100 years and so I've reached 3/4 of that milestone. The only question is how far along the remaining 1/4 will I progress before being cut short.

According to Wolfram Alpha, I have a 50% chance of making it halfway. See Figure 1.

Figure 1

87.5 is the halfway point between 75 and 100. 87.22 is just shy of that. So 50% of my cohort of Australian males will make it to that mark and 50% won't. That's the cold, stark statistic. 25 years is commonly regarded as a generation and so three generations are now behind me. Here is a link to a PDF fact sheet about the number 75 titled Importance Of Number 75 In Mathematics and Other Fields.

Looking at the information about 75 on Numbers Aplenty however, we find more interesting facts. For example, I discovered that it forms a betrothed pair with 48 and that together they form the first such betrothed pair. I'd not heard of this term before but it's defined as follows:

Two numbers \( (m,n) \)  form a betrothed pair if the sum of nontrivial divisors of one number equals the other, i.e., if  \( \sigma(n)-n-1= m\)  and  \(\sigma(m)-m-1 = n\).

The initial pairs are (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128), (8892, 16587), (9504, 20735), (62744, 75495), (186615, 206504).

The same source informed me that 75 is a repfigit number defined as follows:

Let  \(n\)  be a number with  \(k\)  digits. Let us define a Fibonacci-like sequence 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.

The term repfigit is short for repetitive Fibonacci-like digit and such numbers are also named Keith numbers (Wikipedia link).

For example, 1104 is a repfigit or Keith number because the resulting sequence 1, 1, 0, 4, 6, 11, 21, 42, 80, 154, 297, 573, 1104, contains 1104.

Note that the 6 repfigit numbers with 2 digits are, by definition, fibodiv numbers, too.

The first repfigit numbers are 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284 

See my blog post titled Fibodiv Numbers to find out what they are about. In the case of 75, a two digit number, we have 7, 5, 12, 17, 29, 46, 75 and thus it qualifies.

75 is also a trimorphic number defined as a number \(n\) such that \(n^3\) ends in \(n\). Thus we have:$$75^3=421875$$The initial trimorphic numbers are: 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999. It can be noted that 76 is also trimorphic:$$76^3=438976$$My age in days is 27394 which factorises to 2 x 13697 and thus my life can be divided into exactly two halves, each of length 13697 days. I turned this number of days old on October 3rd 1986. The number 27394 has the property that it is equal to 163 x 167 + 173 where 163, 167 and 173 are successive primes. The initial numbers with this property are:

11, 22, 46, 90, 160, 240, 346, 466, 698, 936, 1188, 1560, 1810, 2074, 2550, 3188, 3666, 4158, 4830, 5262, 5850, 6646, 7484, 8734, 9900, 10510, 11130, 11776, 12444, 14482, 16774, 18086, 19192, 20862, 22656, 23870, 25758, 27394, 29070, 31148, 32590, 34764, 37060, 38220, 39414, 42212

These numbers form part of OEIS A292926.

Fibodiv Numbers

My diurnal age today is 27321 and I was struggling to find an interesting sequence to which this number belonged. Fortunately, Numbers Aplenty came to my aid with the information that 27321 is a fibodiv number. Such numbers are few and far between. Here's the definition that is provided by the described source:

These are numbers \(n\)  whose representation can be split into two numbers, say \(a\) and  \(b\), such that the Fibonacci-like sequence which uses \(a\) and \(b\) as seeds contains \(n\)  itself.

 In the case of 27321, it can be seen that this is indeed the case:

273, 21, 294, 315, 609, 924, 1533, 2457, 3990, 6447, 10437, 16884, 27321

The sequence of such numbers can be found in the OEIS A130792 but they are not referred to as fibodiv numbers. 

A130792

Numbers \(n\) whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing \(n\) itself.

The initial members are:

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 

Between 20,000 and 40,000, the numbers are:

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

As can be seen, there's quite a gap between 27321 and the previous sequence member, 26020, but the subsequent member, 27483, is much closer. All the numbers in the sequence admit of only one concatenation and it is not known if there are numbers that admit of more than one.

The sequence is infinite since 19 with seeds 1 and 9, 199 with seeds 1 and 99, 1999 with seeds 1 and 999, 19999 with seeds 1 and 9999 and so on are in the sequence. For example:

1, 9999, 10000, 19999

It's easy enough to confirm that a number is a fibodiv (see permalink) but finding them initially is more challenging. However, OEIS comments list a method but I don't quite understand it.