Perfect numbers
The next two
perfect numbers, and the only other ones known in ancient Greece were 496 and
8128. One problem, which I believe still has to be solved, is to find an odd
number which is also a perfect number. This doesn’t have much to do with
anything, but there are a few things in this chapter which look like that at
first. Keep watching!
Abundant numbers
An abundant number is a number for which the sum of divisors (including the number itself) is more than twice that number, or putting it another way, the sum of proper divisors is greater than the number. So 12 has the divisors 1, 2, 3, 4, 6, 12, so the sum of the divisors is 28 so 12 has an abuncance of 4. The proper divisors (1, 2, 3, 4, 6) sum to 16, so an abundance of 4, also.
The first twenty abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, all of which are even.
The smallest odd abundant number is 945, and there are a few others: 945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175...
The 175th odd abundant number is 81081, and this is the first odd abundant number not divisible by 5, but is it the first odd abundant number not ending in 5? A bit of web searching leads me to believe that it is, but be very wary of AI summaries on this topic. While fact-checking, I came across this nonsense from AI:
153 is abundant because the sum of its proper divisors (1, 3, 9, 17, 51) is 72, which is greater than 153.
So ignore AI. As a small boy, I used to calculate the powers of 2 in my head: 2. 4. 8. 16... to put myself to sleep. Later, I still dull my mind in the dentist's chair by working out the cube root of a random two-digit prime number. It seems to me that finding interesting abundant numbers could fill both the sleep-inducing and pain-relieving roles.
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