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.
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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 repdigit 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
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