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.

Keith Numbers

A Keith Number is an \(n\)-digit integer N with the following property: If a Fibonacci-like sequence (in which each term in the sequence is the sum of the \(n\) previous terms) is formed, with the first \(n\) terms being the decimal digits of the number N, then N itself occurs as a term in the sequence. For example, 197 is a Keith number since it generates the sequence (source)

1, 9, 7, 17, 33, 57, 107, 197, ...

The Keith numbers form OEIS A007629 and the initial members are as follows:

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, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607

Let's take 31331 as another example (permalink):

3, 1, 3, 3, 1, 11, 19, 37, 71, 139, 277, 543, 1067, 2097, 4123, 8107, 15937, 31331

I encountered a variation of Keith numbers when I celebrated being 27223 days old. One of the properties of this number is that it's a member of OEIS A274770:

A274770

Cube analog to Keith numbers.

Here are the OEIS comments regarding this sequence:

Like Keith numbers but starting from \(n^3\) digits to reach \(n\). Consider the digits of the cube of a number \(n\) . Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

The example of \( 776^3 = 467288576 \) is given:

4 + 6 + 7 + 2 + 8 + 8 + 5 + 7 + 6 = 53

6 + 7 + 2 + 8 + 8 + 5 + 7 + 6 + 53 = 102

7 + 2 + 8 + 8 + 5 + 7 + 6 + 53 + 102 = 198

2 + 8 + 8 + 5 + 7 + 6 + 53 + 102 + 198 = 389

8 + 8 + 5 + 7 + 6 + 53 + 102 + 198 + 389 = 776

The initial members of the sequence are:

1, 8, 17, 18, 26, 27, 44, 55, 63, 80, 105, 187, 326, 776, 1095, 2196, 6338, 13031, 13131, 25562, 27223, 70825, 140791, 553076, 632489, 1402680, 1404312, 3183253, 11311424, 50783292, 51231313, 182252596, 255246098, 522599548, 1180697763, 2025114819, 2137581414

In a similar vein, we have the square analogs of Keith numbers. These numbers comprise OEIS A274769: 

A274769

Square analog to Keith numbers.

The OEIS comments state that:

Like Keith numbers but starting from \(n^2\) digits to reach \(n\). Consider the digits of the square of a number \(n\). Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

 The example of \( 1264^2 = 1597696 \) is given :

1 + 5 + 9 + 7 + 6 + 9 + 6 = 43

5 + 9 + 7 + 6 + 9 + 6 + 43 = 85

9 + 7 + 6 + 9 + 6 + 43 + 85 = 165

7 + 6 + 9 + 6 + 43 + 85 + 165 = 321

6 + 9 + 6 + 43 + 85 + 165 + 321 = 635

9 + 6 + 43 + 85 + 165 + 321 + 635 = 1264

The initial members of this sequence are:

1, 9, 37, 40, 43, 62, 70, 74, 160, 1264, 1952, 2847, 12799, 16368, 16584, 42696, 83793, 97415, 182011, 352401, 889871, 925356, 1868971, 1881643, 3661621, 7645852, 15033350, 21655382, 63288912, 88192007, 158924174, 381693521, 792090500, 2025078249, 2539401141

Finally, let's return to Keith numbers and include some further comments from Mr Keith himself (source):

There are still a number of unanswered questions about these numbers, such as:

Are there an infinite number of Keith numbers? Heuristic arguments, and the numerical evidence above, both strongly suggest that the answer is "yes" - in fact, we expect to find roughly \(0.9 \log_2{10}\) (about 3) of them between each power of 10. But there is still no proof, constructive or otherwise, that there are an infinite number of them.

Define a cluster of Keith numbers as a set of two or more (all with the same number of digits) in which all the numbers are integer multiples of the smallest one in the set. There are only three known clusters: (14, 28), (1104, 2208), and the remarkable - for having three members - (31331, 62662, 93993). Question: is the number of Keith clusters finite or infinite? Not only do we conjecture it is finite, but we conjecture that the above three clusters are the only ones.  But we have no clue how to prove this.

Is \(n=10\) the only number of digits for which there are no Keith numbers?  (We tentatively think not, but it may be a while before another one is found.)