Numbers or Words: It Doesn’t Matter

This is my third and final part in a 3-part series on memorization. Two weeks ago, I posted the first part, Memory: Use It Or Lose It, in which I described how one can memorize a list of words. This could be a grocery list, a roster of names, or a vocabulary list. By using a “memory palace”, or a physical place you remember from your childhood or recent past, and bizarre and crazy visualization techniques, you can memorize very long lists!

Last week, in my post It Does Not Take A Math Person To Memorize Numbers, I described how one can use a mnemonic that associates each of our 9 digits to specific sound in our language to then transform long sequences of numerical digits to a much shorter sequence of words. This, in turn, by using methods of part 1, can then be much easily memorized than the long sequence of numbers themselves.

Finally, we put the two together. I’m going to use a small portion of the periodic table of elements from chemistry as an example. Here are the first 10 elements.

  1. Hydrogen
  2. Helium
  3. Lithium
  4. Beryllium
  5. Boron
  6. Carbon
  7. Nitrogen
  8. Oxygen
  9. Fluorine
  10. Neon

Using Pegs

Although we could memorize this list just as we did with the memory palace and visualization technique from the first post, if we wanted to know what the 10th element was, we would need to walk through our palace counting all the way up until we arrived at the 10th element in our list.

There is a way to go directly from 10 to Neon, or from Neon to 10, without having to walk through your palace, and this is done using pegs.

Again, let’s get a place in our minds that we know very well. It could be a relative or friend’s house, a previous work place, or a previous place you have lived.  What can we do to the outside that would make you arrive here in your mind if the periodic table of elements came up?  Perhaps a crazy looking lab scientist or a huge sign on the front of the place that is the periodic table of elements would work.

Most of us know that hydrogen and helium are the first two elements and don’t really need a visualization for these two, but I will proceed as if we they were not readily available in your mind.

  1. Hydrogen sounds like “hide roach in”. Visualize a really big, disgusting, creepy-crawly cockroach and a hat big enough to hide that same roach in.  The hat is important, as it will be your peg for 1.  Place this hat in the driveway/parking lot of the place in your mind. And hide the roach in it.
  2. Helium sounds like “heal Liam”, so I would visualize a hen in doctor’s attire with a stethoscope around its neck, bending over the sick body of Liam Neeson. This can be taking place between the house and the entrance, or just at the entrance of the place in your mind. Why a hen? That is a peg for 2.
  3. For lithium, visualize a small model of your current home (even if you happen to be using your current home as the place in your mind), and any person you would like sitting next to it holding out their thumb that is on fire.  This will hopefully get you to “lit thumb” and the peg word home, which is equal to 3 in the mnemonic major system.
  4. Now visualize the person with the lit thumb yielding a bow and arrow, and shooting the arrow at a bear that is charging out of a neighboring room.  It falls and rolls.  You roll the bear some more.  “Bear, roll him” will hopefully get you to Beryllium, while the arrow is the peg for 4.
  5. Within the next room, visualize Borat (played by the actor Sacha Baron Cohen) riding on top of a whale.  Borat should get you the element Boron, and the whale will be the peg for the number 5.
  6. Rolling down the hallway, you see the Oscar Meier wienermobile but without the wiener run over a big shoe.  The hot dog-less vehicle is just a car bun, which will get you to the element Carbon, while the shoe that it runs over is a peg word for 6.
  7. Traveling through the place we have in our mind, the next sight we see (visualize) is a cow taking a pull off of a special Nitrogen tap (sometimes beer is infused with nitrogen instead of CO2 as it gives the beer a creamier mouthfeel) of something dark and delicious. Perhaps a Guinness?  The element is obviously Nitrogen, but the cow is the peg for 7.
  8. Next up we have a horse with a oddly shaped hoof. It has tubes around its nose and is attached to an oxygen tank (maybe it has smoked way too much and that is what has caused the crazy burn holes in his hoof.  Oxygen is the element, hoof is the peg for 8.
  9. Go to another spot in the place in your mind and find a pie made of toothpaste. Most toothpastes have fluoride. Visualize a bunch of empty toothpaste tubes around a pie. You may want to also visualize a string of toothpaste in the shape of the letter N on top of the pie to remind you that the element is Fluorine rather than fluoride. The pie is a peg word for 9.
  10. In the next place you pass through or by in your mind palace will be a huge neon sign of a daisy.  The sign will give you the element Neon, and its shape (a daisy) is a peg word for 10.

Are you on a roll?  Keep going by visualizing something for Sodium and the peg word tattoo for 11. Then, create something for Magnesium and the peg word tuna for 12.

You can come up with your own peg word if you don’t like the ones that are given on the Wikipedia page for the Mnemonic Major System.

This memory palace is a little different than the one we learned in the first post, as you will be able to do much more than simply recall the list from beginning to end.

For example, if you were to recall the fifth element, you would need to remember that the peg word for 5 is whale, and we can then retrieve from our memory the fact that Borat is riding that whale which reminds us of the element Boron.

Or, which element number is nitrogen?  Well, we remember a cow drawing a beer from a nitrogen tap, and cow is a peg word for 7.  This process of memorization saves you from having to go through an entire list!

Save our future generations, and start using your memory!

It Does Not Take A Math Person to Memorize Numbers

Using random.org, I generated the following 100 random digits.

0	7	2	4	9	7	2	2	5	2
5	1	3	0	7	5	7	7	9	4
1	9	0	6	4	8	2	4	7	1
1	1	4	7	0	8	9	5	7	1
4	9	6	5	6	8	2	4	4	4
4	8	3	1	3	7	0	1	8	4
0	3	0	3	1	1	0	7	6	1
0	3	2	6	2	4	6	7	7	5
3	7	7	4	1	6	2	5	2	1
5	3	2	9	4	6	6	3	9	0

This is part 2 in a 3 part series on memory and memorization. If you missed Memory: Use It Or Lose It, you’ll want to start with that post since we’ll be using the tools discussed there.

The Mneumonic

First, we must develop a way to associate each of the 10 digits from 0 to 9 with a sound. The Memory Book by Harry Lorayne and Jerry Lucas does this very well. For a quick reference , consult the Mnemonic Major System on Wikipedia. I will spell this association and mnemonic out in my own words below.

0. Zero begins with the letter z. When you make the z sound, notice that your mouth is also in the same position it would be to make an ‘s’ sound (or a soft c, or an x as it is used in xylophone).

1. There is one vertical stroke used in the letters t and d (even the uppercase T and D).  Notice when making these sounds, your mouth is in the same general position. This is where we also place the th sound even though the mouth position is a little different.

2. There are two vertical strokes used in the letter n (and uppercase N). If you are familiar with the sign language alphabet, you use two fingers draped over your thumb to represent the letter n.

3. There are three vertical strokes used in the letter m (and uppercase M).  Again, the sign language alphabet has you use three fingers draped over your thumb to represent m.

4. The number four spelled out ends with the letter r.

5. Holding all five fingers of your left hand up in front of you (palm outward), your forefinger and thumb are in the shape of an L.  Learning from Wikipedia just today, you can also use the fact that L represents 50 in Roman numerals.

6. When writing the letter J, if you continue the curve, you can finish creating a backward 6.  Again, learning from Wikipedia, the letter G looks a lot like a 6.  Making the J sound with your mouth, it is also in the position to make a soft G sound, as well as a CH (in chef or cheese) or SH sound.

7. You can put two sevens, point to point and create what looks like a K, or k. Notice that the hard c and the hard g requires your mouth to be in the same position as it would be to make the k sound.

8. When you write an f in cursive, you create a figure 8.  When making the f sound, notice your mouth position does not have to change much to make the ph, gh (as in tough), or v sounds.

9. Flip the 9 around a vertical axis to create a P.  Rotate it 180 degrees to create a lowercase b. Both the p and b sounds require the same mouth positions.

You ignore vowel sounds, along with h, w, y. Once you get used to the system, you’ll also learn to ignore silent letters like the d in judge or the t in patch.  Judge would simply code as 66 and patch as 96.  Let your mouth do the work for you. Since you don’t pronounce both the m’s in hammer, it codes as 34, not 334.

Getting these digits associated with all those sounds takes a little practice.  Try playing on the Major System Database by typing in several words of different lengths to get the hang of it. Also, try typing in numbers with 3, 4, and 5 digits to see what words they can come up with that match those digit lengths.

Personally, I use a slightly different version than the system on the link in the previous paragraph. For example, for the word “fix”, I would pronounce this as “fiks” and code it 870 (8 for the f sound, 7 for the k sound, and 0 for the s sound).  This website has decided to ignore this x sound!

Memorizing 100 Random Digits

In order to memorize any number of digits you now have a system that can change it into words. You cannot expect to be able to come up with nice, sensible sentences with just any sequence of numbers. However, if you come up with several weird and interesting words, you can use the tools used in the previous post to memorize those.

Let’s look at the following 10 bizarre “sentences.”

  1. Scan her bikini online.
  2. Healthy mosaic Hulk goober.
  3. Dubious usher of Norway God.
  4. Tow truck safe bulkhead.
  5. Rope shall chaff anywhere rear.
  6. Horrify Madame Cousteau of Eire.
  7. Awesome, somewhat hideous, hog shit.
  8. Semen. January choke glue.
  9. Make quart Chanel nude.
  10. Lawman approach jumbo ass.

Once you are used to the major system mnemonic, you can then use words to help memorize long strings of numbers.  Each of these “sentences” become strings of 10 digits, all together forming the 100 digits at the opening of the post.

I’m sure you would agree that memorizing those 10 “sentences” (using visualization) would be much easier than memorizing a string of 100 digits.  Of course, when first learning, it takes a little while to understand and incorporate the mnemonic.

How the Pros Do It

Those that are required to memorize long sequences of digits as part of their job or simply for competition (such as the USA Memory Championships) do so by having a set of 2-digit pegs memorized.

That is, they already have a noun, verb, and an adjective on hand for the 100 two-digit pegs from 00-99. Look at the middle of the Mnemonic Major System Wikipedia page for all 300 of these 2-digit pegs. For example, the noun, verb, and adjective for the digits 42 are urine, ruin, and runny.  (The r sound followed by an n sound).

Let me illustrate how the professional would memorize the first 10 digits of the 100 that were given above. That is, the digits 0724972252.

They would see this as 07 24 97 22 52 and be able to retrieve pegs in the order adjective, noun, verb, adjective, noun. In this case, using the linked web page above, they would almost instantaneously retrieve the pegs “sick winery poke neon lion” and concentrate on whatever a sick winery poking a neon lion would look like in their heads.

Since speed has never been necessary for any kind of memorization of numbers that I encounter, I’m not at the level of using pegs. Instead, I just take a little bit more time and come up with even crazier words and sentences like those that you found in the previous section. It makes the visualization a little easier.

I’m soon off to Decorah, IA for a biking and kayaking trip with many friends. I’ve memorized all of their phone numbers as a party trick, but having at least one of them in my memory is a good safe guard.

Memory: Use It Or Lose It

When I was attending Iowa State University, I took a class in Religious Studies with Dr. Hector Avalos. On the first day of class, he recited the roster from memory. I remember thinking how amazing that was.

Then, in 2011, after watching Limitless with Bradley Cooper and understanding that we cannot simply take a pill to have limitless memory and that it takes work, I picked up The Memory Book by Harry Lorayne and Jerry Lucas.

The feat of memorizing an entire roster had always stuck with me, and so I finally put the techniques I learned in The Memory Book to practice while I was at Truman. Memorizing all of my student’s names on the first day of class was still very challenging, but I put the necessary time in.

Classes are beginning at Washburn, and the time has come once again for me to bring back those memorization skills. In a timely fashion, I happened to be listening to the audio book Moonwalking with Einstein: The Art and Science of Remembering Everything by Joshua Foer.

This book reminded me of the tools I picked up in The Memory Book, such as visualizing ridiculous things in order to memorize a list or roster.  It also provided a great example of using a memory palace.  While I have always used visualizing ridiculous things in sequence to memorize a roster, I had yet to employ the use of the memory palace. I gave it a whirl, and was able to memorize the roster of 30 names from my first section of statistics in about 30 minutes.

Whoa! I thought. That was the fastest I think I’ve ever done that!

For the second section of 30 names, I decided to time myself. I stopped the stopwatch at 21 minutes and 7 seconds and was able to recite the entire 30 names.

Three Part Series on Memory

This post will be part 1 of a three part series. In this part, we focus on memorizing a list of words. The words we will happen to memorize are names, but they could be a grocery list, a to-do list, or simply a list of random words.  It would not matter.

Part 2 will give you tools of how to memorize long strings of numbers.

Part 3 will combine the two parts and provide a method that will help you memorize not only a word, but a number that is associated with it. This combination of techniques might be good in memorizing all the U.S. Presidents or the periodic table of elements in such a way that it wouldn’t matter if you were asked any of the following:

  • Which president (number) was James Polk?
  • Who was the 19th U.S. President?
  • What is the atomic number of Argon?
  • What is the element with atomic number 24?

Combining the techniques will provide you with a way to answer either type of question with relative simplicity. Let’s first start with part 1, however, which is memorizing a list of names (or words).

How to Memorize a List

Think of a house in which you are very familiar, and in particular, one that has several rooms and a driveway. I will walk you through how I would memorize the following list of 12 names which I randomly generated using Behind the Name.

  1. Hillary Bunker
  2. Roni Coy
  3. Brion Derby
  4. Steven Garbutt
  5. Fulk Jackson
  6. Jade Johnson
  7. Tillie Millhouse
  8. Earnestine Morrish
  9. Lynnette Pender
  10. Moreen Petit
  11. Angelica Quincey
  12. Lila Sheppard

First, I like to read through the names a few times to get familiar with them, looking for names that I may know well. For example, Hillary, Brion (the name, not the spelling), Steven, Jackson, Johnson, and Sheppard are all familiar to me and I can come up with familiar faces for each of these names.

Next, you’ll need to be able to visualize standing at the end of the driveway or walkway to the house facing the front door. What you will be doing is walking up to the front door, going inside, and doing a small tour of the house in your mind. Along the way, I will insert some people or items doing crazy things in different locations that should not make any sense.

Take your time after reading each step below to close your eyes and get a vivid image of not only what is being described, but also the movement you make through the house.

  1. Here we are at the end of the driveway of the house looking toward the front door. Imagine the former first lady, Hillary Clinton, has dug herself a bunker there.  Really get that image burned into your mind. In fact, the bunker is dug in such a way that it leads to the front door.
  2. Now, imagine the bunker fills with water and Hillary gets into a boat and rows herself to the front door, with each row hitting her knee. A bunch of decoy ducks are all over the water.
  3. You probably know someone named Brian. Do you happen to know the actor Bryan Cranston (from Breaking Bad)?  Anyway, think of a Brian/Bryan/Brion opening the front door to welcome you home, but they are all dressed up as if they are going to the Kentucky Derby as a woman.  (Think of those crazy hats).
  4. Since I don’t know the house you are using very well, you will be somewhat on your own as you navigate through the house.  The rest of the images could be in a hallway, a staircase, or inside one of the several rooms. Let’s put a huge picture of Stephen Colbert (or, if you don’t know who that is, choose a Steve you know) on a nearby wall right inside the front door.  Now visualize him mooning you in the picture with big black letters GAR printed across his butt.
  5. Where are you now in the house?  Let’s place Michael Jackson there alongside Spock. Jackson looks at Spock, and in his very soft voice, asks, “Who the Vulc are you?” (Spock is a Vulcan).
  6. Whether there is a television there or not, place one near Michael and Spock.  Playing on the TV is this scene from The Big Lebowski: The Dude: “F*** sympothy! I don’t need your f***in’ sympathy, man, I need my f***ing johnson!”  On top of the TV is the ace of spades with a big letter J on it to remind you that it isn’t a spade, it is a Jade. Make the card really big so you won’t accidentally miss it later.
  7. Let’s now move to the next room. Perhaps you’re familiar with the character Milhouse from The Simpsons?  If so, visualize him with a garden till, tilling away at the floor. Or maybe you can visualize a grain mill a little easier? As a homebrewer, I can. Maybe put this mill inside a large doll house so that it is now a mill house. Imaging trying to mill a doll-sized garden till. Either one should work.
  8. Remember Morris the Cat in the 9 Lives commercials back in the 80’s? He’s a fat orange tabby with a deep goofy voice.  Visualize this cat sitting around the corner from the room that you are in. As you are traveling to the next room, there he is! Imagine he sits next to a beer stein, and with a lisp, says “Hi, I’m Morrish. Want to earn a stein? Just rub my belly.”  Visualize the entire process. He sits there. He speaks. He rolls over. You rub his belly. You earn the stein, so you take it.  Now, walk into the room you were headed to.
  9. In this room there are a bunch of loons. All of them are waddling around saying “Der… Der… Der.”  All but one are penned up.  You throw a net around the one that is not in the pen.  You just threw a loon net. (Hope this gets you to Lynnette).  Now, throw it back in the pen. Why are all these loons saying “Der“?
  10. Along the way to the next room, you see your pet in distress (think of a former pet if you don’t currently have one).  Perhaps it is Morris from a memory ago? It is yelping in pain.  Luckily, there is some morphine available right next to your pet in a huge bottle with MORPHINE written on the side (the middle letters scratched out).  You give your pet some morphine and pet it.
  11. You hear the song “Hallelujah” being sung in the next room. Upon entering you see Angels singing. They are singing along with Quincy Jones (click on the link if you are not familiar with this musician). They stop singing and the Angels lick Quincy.
  12. Jealous of all the licking, a singing German Shepherd wearing Lee jeans runs in singing “la la la la la la la” before getting its own lick in.

What craziness did you just read?

Scroll down now so that you can hide the names and the instructions.

No peaking!

OK, now. In your mind, let’s get back to the end of your driveway looking back at your house.  Can you recite the 12 names?

If you failed, it is probably because you did not get the image you were supposed to visualize vivid enough. Another review of the instructions and you can probably get it.

Obviously, the work isn’t the memorization part, but it is how creative and imaginative you can be. In this world of smartphones, memory is becoming a lost art. Memorizing a roster 40 years ago would not have been that big a deal. A few hundred years ago, memorizing the list of names may have been the only method of getting the roster!

Now, memorizing a roster is a parlor trick. A few times, I have even received applause! The capacity of our memory is fading. What does this mean for future generations?

Perhaps the part of the brain we use for memorization isn’t really a “part” and is just a process that will no longer be needed. Other “processes” will take memorization’s place.  This is looking at it optimistically.

Perhaps, there is an actual part of the brain used for memorization, and that “part” will just become mushy and dormant. That is the more cynical thought.

I’m pretty sure we still do not know the answer. I only have my own life to live, however, so I’m going to use it rather than lose it.

Aliquot Divisors

When the idea came to write about this post, I had never came across the term “aliquot” to my recollection.  Let’s quickly deconstruct that sentence and assign some probabilities.

  • I’ve never came across the term “aliquot”: probability = 0.01
  • I don’t recall having come across the term “aliquot”: probability = 0.99

From this, I hope you’ve come to understand a few important things.

  • Probabilities must sum to 1
  • All probabilities must be non-negative and less than (or equal to) 1
  • I most likely don’t have a great short term memory.
  • I am most likely a pretty funny dude.

Sum of Divisors Function

After searching the term “Sum of Divisors function” a page with a wiki on the Online Encyclopedia of Integer Sequences came up.  On this, they define a function for the sum of the divisors for a number.

Let’s do a quick example before introducing notation and look at the number 40 (that’s how old I am for another month and a half).  What numbers divide 40 evenly?  That is, if you push 40, and then \div, what numbers can you push next so that the result is an integer (not a decimal).

Here are the divisors of 40: 1, 2, 4, 5, 8, 10, 20, 40.  A function that computes the sum of the divisors of a number should give an output of 1+2+4+5+8+10+20+40 = 90 whenever you input the value 40.

Now for the notation.  The sum of the divisors function is given as \sigma(n).  To apply the above example, this means that \sigma(40)=90. In other words, you input the value 40 into the sum of divisors function \sigma(\cdot), and the output is 90.

Let’s try a few others. What if we input 10?  That is, what is the value of \sigma(10)?

First, let’s find the divisors of 10: 1, 2, 5, 10. Next (and finally), sum them together: 1+2+5+10 = 18.  Thus, \sigma(10) = 18.

How about \sigma(11)?  The only divisors of 11 are 1 and 11 (which, as a side note, means that 11 is a prime number).  This means \sigma(11)=12.  If you inspect this example thoroughly, you should be able to deduce that the sum of the divisors of any prime number will be one more than the number.

Since 5 is a prime number, \sigma(5)= 6 (the sum of 1 and 5).  Likewise, since 7 is a prime number, \sigma(7)=8.

Now, try a few on your own.  Find \sigma(15) and \sigma(24).  The answers are below, but you can also check them by typing “sigma(15)” or “sigma(24)” into WolframAlpha. Technically, the correct code is DivisorSigma[1,15] and DivisorSigma[1,24], but you need not concern yourself with that.

So What are Aliquot Divisors?

The aliquot divisors of a number are what I have always known to be proper divisors of a number, which are all the divisors except the number itself.

So, the aliquot divisors of 40 are 1, 2, 4, 5, 8, 10, and 20 (we leave off 40).  And the sum of the aliquot divisors function is denoted by s(\cdot).  For our example, then, s(40)=1+2+4+5+8+10+20 =50.

With this sum of aliquot divisor function, s(n), we can now explore some fascinating things about different numbers.

Prime Numbers

For every prime number (a number only divisible by 1 and itself), the only aliquot divisor is 1.  So, by plugging any prime number into the function s(\cdot), the result will be 1.

Deficient Numbers

The numbers 4 and 10 are deficient numbers. Why? When you plug them into the sum of aliquot divisors function, s(\cdot), you will get a value smaller than the number.

The aliquot divisors of 4 are 1 and 2.  So, s(4)=3.  The aliquot divisors of 10 are 1, 2, and 5. So, s(10) = 8. Notice that each of the output values are smaller than the input values.  That means the numbers are deficient. 

Let’s recall what you learned about prime numbers in the previous section.  If p is a prime number, than s(p)=1.  From this we can deduce that all prime numbers are deficient.

Furthermore, since we have the above stated examples of 4 and 10 as deficient numbers, and we know they are not prime since they are divisible by something other than themselves and 1, then we can also deduce that not all deficient numbers are prime.

Abundant Numbers

The numbers 12 and 40 are abundant numbers? Why? When you plug them into the sum of aliquot divisors function, s(\cdot), you will get a value larger than the number.

We have already seen that s(40)=50. The aliquot divisors of 12 are 1, 2, 3, 4, and 6.  Thus, s(12) = 1+2+3+4+6=16.  Each output value is larger than the input value. That means the numbers are abundant. 

Perfect Numbers

The numbers 6 and 28 are perfect numbers? Why? When you plug them into the sum of aliquot divisors function s(\cdot), you will get the same value!

The aliquot divisors of 6 are 1, 2 and 3.  Thus s(6)=1+2+3=6.  The aliquot divisors of 28 are 1, 2, 4, 7, and 14.  Thus, s(28) = 1+2+4+7+14 = 28.  Each output value is the same as the input value!  That means the number is perfect.

Amicable Chains

Recently, I was introduced to the concept of amicable chains.  What if we continue to apply the sum of aliquot divisor function s(\cdot) over and over again?  For example, let’s try it with 12.  As we saw in the Abundant Numbers section, s(12)=16.

Now, what if we plug in 16?  Aliquot divisors of 16: 1, 2, 4, and 8.  Thus, s(16)=15.

Again? What if we plug in 15? Aliquot divisors of 15?  1, 3, and 5.  Thus, s(15)=9.

Again? Aliquot divisors of 9? 1 and 3.  Thus s(9)=4. The aliquot divisors of 4 are 1 and 2, so s(4)=3.  Since 3 is prime s(3)=1 and now we’re trapped at 1 forever.

This created the sequence

\displaystyle 12\rightarrow 16\rightarrow 15\rightarrow 9\rightarrow 4\rightarrow 3\rightarrow 1

If you do a lot of exploration, a lot of numbers will produce a sequence that ends at 1 and becomes trapped. Some don’t. Take the perfect numbers 6 and 28 for example.

Since s(6)=6, then 6 has a chain of length 1 since it goes directly back to itself.  So does 28.

There are other numbers that do weirder things.  Take 220 and 284 for example (you will definitely need to use WolframAlpha on these if you want to check. However, you’ll need to type in “sigma(220)-220” and “sigma(284)-284”.)  The sum of the aliquot divisors of 220 is 284, and the sum of the aliquot divisors of 284 is 220.  This produces a chain of length 2. There are several chains like this one.

With the aid of a computer program, I’ve found one of length 5 and another of length 28.  Using a similar program, I also stumbled upon interesting numbers like 1230 and 1248.  Although they do not produce chains, they take a while to get back to 1.  Using a very cool trick of emailing a former colleague with a higher level of number theory understanding, he was able to find out that it takes 185 terms to get back to 1 starting at 1230 and 1076 terms to get back to 1 if you start at 1248.

The interesting thing about these sequences are how really, really, big they get.


Answers to Above Posed Questions

\sigma(15) = 24, and \sigma(24) = 60.