Facts and Curiosities About Abundant Numbers

In ancient times, the mathematician Nicomachus of Gerasa (60 to 120 A.D.) summed the proper divisors of many numbers (all the divisors except the original number) and discovered they could be categorized into three different classes:

1. Abundant numbers - those with a divisor sum larger than the original number. For example, 20 has the divisors (1, 2, 4, 5, 10, 20) and summing the "proper" ones yields 1+2+4+5+10 = 22 > 20, so it is abundant.

2. Deficient numbers - those with a divisor sum smaller than the original number. For example, the divisors of 15 are (1, 3, 5, 15) and 1+3+5 = 9 < 15, so it is deficient.

3. Perfect numbers - those with a divisor sum equal to the original number. For example, the divisors of 28 are (1, 2, 4, 7, 14, 28) and 1+2+4+7+14 = 28, so it is perfect.

In this article we will be dealing primarily with the first class of numbers, the abundants, and listing a few theorems and curiosities related to them. (The proper mathematical definition for an abundant number is, let sigma(n) equal the sum of the divisors of n including the original number as a divisor, then a number is considered abundant if sigma(n) > 2n.)

Here is the sequence of abundant numbers:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, ...

In his book, Introduction to Arithmetic, Nicomachus included florid and occasionally "mystical" prose to describe certain classes of integers. Concerning the abundants, he wrote they were similar to a creature with "too many parts or limbs, with ten tongues, as the poet says, and ten mouths, or with nine lips, or three rows of teeth."

Our first fact about abundant numbers, and one of the most basic, is that every multiple of an abundant number will also be abundant - hence, there are infinitely many of them. To see this, consider the divisors of 114, which are (1, 2, 3, 6, 19, 38, 57, 114) and thus 1+2+3+6+19+38+57 = 126 > 114, making it abundant. Now if we multiply 114 by 2 we get 228, and its divisors are (1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228) and 1+2+3+4+6+12+19+38 +57+76+114 = 332 > 228, also abundant. The proof that multiples of abundant integers will also be abundant is not exceedingly difficult but it will be omitted here for space (and technical) reasons.

Usually if a number has many small prime factors it will be abundant. But to go in the opposite direction, let's look for the least amount of prime factors an abundant can have. A simple computer search reveals that abundants seem to always require at least three prime factors (counting the factors with repetition; for example, 700 is said to have five factors even though only 2, 5, and 7 appear in its factorization: 2*2*5*5*7 = 700). Here is the sequence of abundants that have at least three prime factors (counting repetition):

12, 18, 20, 30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, ...

I conjecture that this sequence is infinite.

There are even abundant values of n such that (n, n + 2, n + 4, n + 6, n + 8) are all abundant numbers. This sequence could be called the "abundant quintuplets," and these are its initial members:

2988, 4728, 9724, 18844, 22984, 30544, 35148, 39948, 45048, 50464, 55788, 56808, 58056, 58780, 69184, 78048, 81948, 85744, 101148, 106144, 108256, 109248, 117124, ...

For example, the first term is 2988 but 2990, 2992, 2994, and 2996 are also abundant. We will leave it up to the reader to find the divisors of these numbers and add them up to check them.

The sequence above is A108926 in the Online Encyclopedia of Integer Sequences.

Looking at the original sequence of abundants near the top of this article, we notice that no odd numbers appear. Are there any odd abundant numbers? Yes. A lot of them.

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, 14805, 15015, 15435, 16065, 16695, 17325, 17955, ...

Claude Gaspar Bachet (1581 to 1638) was the first person to discover the odd abundant, 945. Now they are quite easy to find with a computer search.

Charles de Neuveglise in the year 1700 put forth the conjecture that multiplying together two consecutive numbers n*(n +1) - (this form of integer is called 'oblong') - will always produce an abundant as long as n is greater than 3.

But he was wrong.

The first counterexample is 110 = 10*11, which has the divisors (1, 2, 5, 10, 11, 22, 55, 110) and 1+2+5+10+11+22+55 = 106 < 110, a deficient number. Other counterexamples are

182, 506, 1406, 1892, 2162, 2756, 3422, 3782, 4556, 5402, 6806, 7310, 8930, 9506, 11342, 11990, 14042, 14762, 17030, 17822, 18632, 20306, 21170, 22052, 22952, 24806, 26732, 27722, 29756, 31862, 32942, 36290, 37442, 41006, 42230,

Which makes up sequence A077804 in the Online Encyclopedia of Integer Sequences.

The "abundance" of a number is also an interesting concept and involves the function, A(n) = sigma(n) - 2n. With this function we can find all the classes of Nicomachus's original sum-of-divisor numbers. If A(n) < 0, n will be deficient. If A(n) > 0, n will be abundant. If A(n) = 0, n will be perfect. Here are the first few terms given by A(n) = sigma(n) - 2n:

-1, -1, -2, -1, -4, 0, -6, -1, -5, -2, -10, 4, -12, -4, -6, -1, -16, 3, -18, 2, -10, -8, -22, 12, -19, -10, -14, 0, -28, 12, -30, -1, -18, -14, -22, 19, -36, -16, -22, 10, -40, 12, -42, -4, -12, -20, -46, 28, -41, -7, -30, -6, -52, 12, -38, 8, -34, -26, -58, 48, -60, -28, -22, -1, -46, 12, -66, -10, -42, 4, -70, ...

To illustrate one use of the abundance function, I will switch to perfect numbers for a moment. No one knows whether any odd perfect numbers exist. In fact, this is a famous unsolved problem. Not a single odd perfect number has ever been found for all n< 10³⁰⁰. But perhaps we can use our abundance function A(n) to see just how close odd numbers will get to being perfect (remember we want A(n) to be close to zero).

Below is the sequence of odd n such that their abundance b satisfies -10 b n is A(n) = sigma(n) - 2n. So in other words, the following odd numbers are only "ten away" in either direction from being perfect.

1, 3, 5, 7, 9, 11, 15, 21, 315, 1155, 8925, 32445, 442365, 815634435 ...

For example, taking the 12^th term, 32445; sigma(32445) = 64896 and 32445*2 = 64890 and 64896 - 64890 = 6, which makes the odd number 32445 only six away from perfection: A(32445) = 6.

The sequence above is A077374 in the Online Encyclopedia of Integer Sequences and I conjecture that it is infinite.

Another class of numbers related to the abundance of n are the "quasiperfect" numbers. These are solutions to the equation sigma(n) = 2n +1, or A(n) = 1; and although they have a name, no one knows if any exist! That's right. Mathematicians and programmers have never been able to find a quasiperfect number, nor have they been able to prove there will never be any; although it has been proven that if n is quasiperfect, it must be larger than 10³⁵ and have more than 7 distinct prime factors.

Even though abundant numbers have been known since ancient times, there are still many discoveries to be made about them. Sequences and facts involving abundants are like brilliant gems lying in an abandoned river bed just waiting for some lucky and curious stroller to come along and pluck them out. Perhaps you would like to start strolling for a few abundant curiosities yourself. Good luck with your research.