When UCLA mathematician Terence Tao appeared on The Colbert Report in November 2014, viewers learned that prime numbers can be "sexy"—if they’re six apart, that is, like 5 and 11.
Though sexy may be the English-to-math crossover most likely to elicit laughter from a studio audience, it turns out that many common adjectives take on specialized meanings when applied to numbers. (Note that the numbers dealt with here are positive integers exclusively. “Number” and “positive integer” are therefore used interchangeably.) Here’s an alphabetized selection.
People can’t be amicable all by their lonesomes, and neither can numbers: amicable numbers come in pairs. Two different numbers m and n are amicable if the sum of all the proper divisors of m is n, and vice versa. (A number’s proper divisors are its positive factors other than itself.)
Consider 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which sum to 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, which—presto!—add up to 220. So 220 and 284 are an amicable pair—the smallest pair, in fact. Care to look for the next smallest?
The mathematical definition of aspiring involves something called an aliquot sequence: a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. So if you start with 10, the second term in the sequence is 1+2+5=8, and the third is 1+2+4=7. Convince yourself that the fourth term is 1, and that this is the last term.
Got that? Okay, back to aspiring. A number n is aspiring if its aliquot sequence terminates in a perfect number (see #10 below) but n is not itself perfect. The number 119 is aspiring, but no one knows if 276 is.
You might think of 16 as sweet, but actually a more apt adjective is deficient. Sixteen is divisible by four positive whole numbers other than itself: 1, 2, 4, and 8. Adding these together yields 1+2+4+8=15. Since 15<16, 16 is deficient.
In general, a number n is deficient if the sum of its proper divisors is less than n. The first 10 deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, and 11.
Quick review of binary notation: The only digits are 0 and 1, and place values are base 2. The rightmost place is still the ones place, but the next one to the left is not the tens, but the twos. Then there’s the fours (4=2²), the eights (8=2³), the sixteens (16=24), and so on. Since 29=16+8+4+1, its binary expansion is 11101.
Note that there is an even number of ones in the binary expansion of 29. Numbers with this property are called evil. (Perhaps you thought all of them were?) Other evil numbers include 17, 24, and 39. Can you name another?
It might seem crazy what I’m about to say, but bear with me: 617 is happy.
Here’s why: Square each of 617’s digits and add up the results. 6²=36, 1²=1, 7²=49, and 36+1+49=86. Now square each of 86’s digits and add up those squares. 8²=64 and 6²=36, and 64+36=100. Repeating the process: 1²=1, 0²=0, 0²=0, and 1+0+0=1.
A number is happy, see, if iterating the operation of summing the squares of its digits eventually leads to 1.
You remember pi, right? The ratio of a circle’s circumference to its diameter? Decimal expansion 3.14159 ... ? In case the annual March 14 helping of pi/pie puns hasn’t already cemented the association between this mathematical constant and food, there’s this: Hungry numbers are defined in terms of pi.
The kth hungry number is the smallest number n such that the first k digits of pi appear in the decimal expansion of 2n.
So the first hungry number will be the smallest number n such that 2n contains 3, the first digit of pi. None of 2¹=2, 2²=4, 2³=8, or 24=16 works, but 25=32 does, so 5 is the first hungry number. The second hungry number is 17, because 217=131072, the first two digits of pi. See if you can find the third.
A 2014 survey by British writer Alex Bellos found that, if you’re trying to guess someone’s “favorite” or “lucky” number, 7 is your best bet. Is 7 even lucky, though, as mathematicians use the word?
To see which numbers are lucky, start with the positive odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 ... Delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21 ... The next remaining number is 7, so delete every seventh number. That leaves 1, 3, 7, 9, 13, 15, 21 ... Next delete every ninth number, then every thirteenth ... you get the idea. The lucky numbers are the ones that don’t get nixed.
So 7 is lucky, after all. Is your favorite number?
Are you dating a narcissist? It’s hardly my place to speculate, but whether a given number is narcissistic, that I can answer.
Look at 153. Written in base 10 (can’t hurt to specify after introducing binary in #4 above), 153 has three digits. Raising each of these digits to the number of digits—3—you have 1³=1, 5³=125, and 3³=27. Add 1+125+27, and you get ... 153! Behold: a narcissistic number!
In general, a k-digit number n is narcissistic if it is equal to the sum of the kth powers of its digits.
Recall the definition of evil as it applies to numbers (see #4 above). Odious is, unsurprisingly, related. A number n is odious if it has an odd number of ones in its binary expansion. Take 31, for example: 31=16+8+4+2+1, so the binary expansion of 31 is 11111. One, two, three, four—count ’em five—ones, and five is odd, so 31 is odious. Seems harsh, I know. (Wondering why they’re odious and evil? Look at the first two letters.)
If you’re over 28, you have missed your chance to be perfect. To be a perfect number of years old, that is. A number n is perfect if the sum of its proper divisors is equal to n. So 28 is perfect because its proper divisors are 1, 2, 4, 7, and 14, and 1+2+4+7+14=28. After 6 and 28, the next smallest perfect number is 496.
Recall the definition of another p-word applicable to numbers: prime. A positive integer greater than 1 is prime if it has no positive divisors other than itself and 1. Now consider 196. The only prime factors of 196 are 2 and 7, and both 2²=4 and 7²=49 divide into 196 without remainder. Therefore 196 is powerful.
Defined generally, a number n is powerful if, for every prime p that divides n, p2 also divides n.
A. K. Srinivasan coined the mathematical meaning of the word practical in a 1948 letter to the editor of Current Science. A number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
Let’s see why 12 is practical. The divisors of 12 are 1, 2, 3, 4, 6, and 12. And since 5=1+4, 7=3+4, 8=2+6, 9=3+6, 10=4+6, and 11=1+4+6, 12 passes the test.
Recall from the aspiring entry (see #2) how to form an aliquot sequence. A number is sociable if its aliquot sequence returns to its starting point. The aliquot sequence for 1264460, for instance, is 1264460, 1547860, 1727636, 1305184, 1264460, ... so 1264460 is sociable.
An untouchable number is a positive integer that is not the sum of the proper divisors of any positive integer.
Let’s unpack that. The proper divisors of—to pick any old positive integer—12 are 1, 2, 3, 4, and 6. These add to 1+2+3+4+6=16, so 16 is not untouchable.
So what is? Two. And 5. Also (skipping ahead) 268 and 322. While legendary Hungarian mathematician Paul Erdős proved that there are infinitely many untouchable numbers, no one has managed to establish that 5 is the only odd untouchable, though it is suspected to be.
Denizens of Portland and Austin may worry about the permanence of their towns’ eccentricities, but there’s no need for “Keep 5830 weird” signs.
Five thousand eight hundred thirty is weird—and always will be—because it meets two criteria: (a) it is less than the sum of all of its proper divisors and (b) it is not the sum of any subset of those divisors.
Seventy is also weird. Witness: The proper divisors of 70 are 1, 2, 5, 7, 10, 14, and 35. And while 70 is less than 1+2+5+7+10+14+35=74, no selection of those summands adds to 70.