Have you wondered what numbers are so remarkable that it’s worth mentioning in the list of extraordinary numbers? Feel free to comment below. Meanwhile, here are 15 of them. Let’s find out what they are.
12 is the LCM of all integers between 1 to 4. Since its prime factorization is 22 * 3, its sum of factors is 7 * 4 = 28, which is greater than twice itself. In fact, it’s the first integer like this, called an abundant number. Remarkably, it’s also the first of the two known sublime numbers, whose number and sum of factors are all perfect. Besides, the 12th Fibonacci number is 144, whose square root is precisely 12.
The 10th term of the Fibonacci sequence is also the 10th triangular number. In fact, this is the only intersection like this after 1. The Fibonacci sequence grows much more quickly than the triangular numbers after the fourth term. You can see that the 11th term of the Fibonacci sequence is 89, while the 11th triangular number is only 66.
Firstly, 6 is the sum and product of 1, 2, and 3 simultaneously. Given that its prime factorization is 2 * 3, its sum of factors is 3 * 4, which is 12. This indicates that 6 is the first perfect number and is associated with the Mersenne prime 3. It is also the first friendly number, whose ratio between its sum of factors and itself (the abundancy index) is shared by another number, which is 28, 496, 8128, and so on in this case.
31 is a Mersenne prime (25 – 1), and it is also a Mersenne prime exponent which produces a prime (231 – 1 = 2147483647), which makes 2147483647 a double Mersenne prime. Besides, the 31st prime is 127, which is another Mersenne prime. More remarkably, it’s the sum of factors of two consecutive square numbers (16 and 25), and it’s a repunit in two different bases (11111 in base 2 and 111 in base 5).
4 = 2 * 2 = 2 ^ 2 = 2 + 2. Moreover, 4 is the first composite number because it’s the square of the smallest prime (2).
This number may only look like a multiple of 9, but it’s very close to being perfect. Its prime factorization is 32 * 72 * 112 * 132 * 192 * 61. However, if you take 192 * 61 = 22021 as prime, the sum of factors will look like this: 13 * 57 * 133 * 183 * 22022 = 397171152378, which is exactly twice 198585576189. But don’t forget that 22021 is composite, so this “spoof perfect number” is abundant. In fact, it is the only positive odd “spoof perfect number”.
650 is barely abundant. Its sum of factors is 1302 and is just two from rendering 650 a perfect number. But this number still stands out, even though there are multiple numbers whose abundance is 2.
Most integers with abundance = 2 are of the form 2n-1 * 2n-3. For instance, 20 is 23-1 * 23 – 3, 104 is 24-1 * 24 – 3, and 464 is also of this form. But 650 breaks this “rule” in an extraordinary way: It is not even divisible by 4. Instead, it is 2 * 52 * 13, whose factors sum up to 3 * 31 * 14 = 1302. According to data from the associated OEIS list, it is. the only rule-breaker of this type until at least 144115187270549504.
3 is the first Mersenne prime and the first Fermat prime at the same time, and is the only number (n), in which n+1 and n-1 are all powers of 2. Besides, it is the start of the longest “chain” of Mersenne primes (22 – 1 = 3, 23 – 1 = 7, 27 – 1 = 127, 2127 – 1 = 170141183460469231731687303715884105727), which has a length of four. The only other “chain” has a length of 2 and comprises of 31 and 2147483647.
351351 is an odd abundant number with an abundance of just 18. But there’s more to it, and first we need to learn about weird numbers.
A weird number is an abundant integer that is not the sum of any subset of its proper divisors. For instance, 70 is abundant, with its sum of factors being 144. Try to sum up the proper divisors of 70 (1, 2, 5, 7, 10, 14, 35) yourself so that you can get 70. But no matter what combination you choose, you cannot obtain 4. Hence, 70 is “weird”.
No odd weird numbers are currently found, but there’s a number that’s very close to being one: 351351. Its abundance is only 18, which can be obtained by summing up 7 and 11. But no matter how you sum up its proper divisors under 18 (1, 3, 7, 9, 11, 13), you cannot obtain 18 without excluding 1. This makes 351351 the only odd number found to satisfy this property: It cannot be semiperfect if you exclude 1 (or include, if you’re using a different method) from its proper divisors.
15 is the product of the first 2 Fermat primes (3, 5) and the first 2 odd primes (3, 5) and has more factors (4) than any odd numbers smaller than 15. But there’s more to it. It is closely associated with 3-perfect and 4-perfect numbers.
The sum of factors of the sum of factors of 15 is 60, which is exactly 4 * 15. But since 15 and its sum of factors, 24, are not relatively prime, 15 * 24 = 360 is not a 4-perfect number. Nevertheless, it is also part of the 3-perfect number 120, which is 8 * 15. The sum of factors of the sum of factors of 8 is 24, which is 3 times 8. Amazingly, 360 is the sum of factors of 120.
2520 is the LCM of all integers from 1 to 10, and it has so many factors that the smallest number whose number of factors is greater than that of 2520 is 5040, which is 2 * 2520. Moreover, 2520 is a factor of every highly composite number greater than 2520. In fact, this is the greatest number that satisfies any of these two properties.
The first Mersenne prime is 22 – 1 = 3, the second Mersenne prime is 23 – 1 = 7, the third Mersenne prime is 25 – 1 = 31, and the fourth one is 27 – 1 = 127. It seems like that, for any prime p, 2p – 1 will be prime. But that’s not true, and the first counterexample is 2047. Although it is 211 – 1, it is also 23 * 89, marking the first composite Mersenne number with a prime exponent.
The Fibonacci numbers are having this issue, too. F2 = 1, F3 = 2, F5 = 5, F7 = 13, and so on. It seems that whenever you get a prime index for a Fibonacci number, you won’t get a composite number. However, this is also false. The 19th Fibonacci number is 4181, which is 37 * 113. Like 2047, this number starts the trend where Fibonacci primes are extremely rare.
This is also the same type of counterexamples as said before, but it’s much more extreme. Take a look at the Fermat numbers, which are integers of the form of 22^n + 1, where n is an integer. The first five Fermat numbers are 3, 5, 17, 257, and 65537, which are all prime. Does this imply that all Fermat numbers are prime? No, it doesn’t. A counterexample is right around the corner, which is 22^5 + 1. 4294967297 is 641 * 6700417, which means that it is not prime.
After 65537, no other Fermat primes have been found, which means that 4294967297 marks the start of a long sequence of consecutive composite Fermat numbers.
1 is in many lists as a trivial item, from the polygonal numbers to the Fibonacci sequence, factorials, highly composite numbers, perfect powers, and so on. But most importantly, it is the only number that is neither composite nor prime because it only has 1 factor. Why is it declared not prime? This is because if 1 were defined as prime, many laws about prime numbers will get much less straightforward and more questionable as it would be otherwise. For instance, if 1 were prime, there would be an infinite amount of prime factorization for every single number and an infinite list for the sum of factors for every number. Prime numbers have exactly two factors, and 1 hasn’t even reached this threshold for minimal divisibility.
If you have a favorite number from this list (or even numbers outside this article), feel free to comment below. Meanwhile, appreciate how extraordinary properties of integers, or even combinations of them, exist in the Universe.
References and Credits
- Evelyn Lamb. (2019, April 2). Why Isn’t 1 a Prime Number? Retrieved August 31, 2021, from https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
- (2020, September 10). Mathematicians Open a New Front on an Ancient Number Problem. Retrieved August 31, 2021, from https://www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/
- (n.d.). A122036 – OEIS. Retrieved August 31, 2021, from https://oeis.org/A122036
- (n.d.). A077586 – OEIS. Retrieved August 31, 2021, from https://oeis.org/A077586
- (n.d.). A088831 – OEIS. Retrieved August 31, 2021, from https://oeis.org/A088831
- (n.d.). A106037 – OEIS. Retrieved August 31, 2021, from https://oeis.org/A106037
- (n.d.). A072938 – OEIS. Retrieved August 31, 2021, from https://oeis.org/A072938