It was Jean's birthday last week. When I know someone's age I sometimes send a few examples of where the number comes up. There was the beautiful Orion nebula M42, a musical based on a street in New York, the atomic number of molybdenum, the critical angle that describes where a rainbow appears and, of course the answer to the ultimate question of life, the Universe and everything.1 Jean liked the rainbow as anyone should, but today it occurred to me that 42 was a term in an interesting sequence.
In number theory there is a set of numbers where the sum of the inverses of the prime divisors plus the inverse of the number is equal to 1. Not many numbers have this property, but a few do
2 the prime factor is 2: 1/2 + 1/2 = 1
6 the prime factors are 2 and 3: 1/2 +1/3 +1/6 = 1
42 the prime factors are 2, 3, and 7: 1/2 + 1/3 + 1/7 + 1/42 = 1
The next few terms are 1806 and 47058 which you can verify for yourself, but after that the numbers grow rapidly.
Neil Sloane is a friend from the Bell Labs days who loves integer sequences - as a passion he built and maintains the world's archive of them - OEIS. I typed in a few terms and there was my sequence - primary pseudoperfect numbers. So Jean - your age is the third primary pseudoperfect number. Possibly a catchy tshirt?
There are an infinite number of integer sequences - some are incredibly useful and others incredibly obscure. Math doesn't have a requirement that sometime be useful. It is nice if it is beautiful and deep, but application to the world is for others to worry about.
If you hear the phrase mathematical sequence you probably think about the Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on. The sequence is defined by Fn = Fn-1 + Fn-2 where F1 = 1 and F2 =1.2 There are any number of beautiful mathematical properties associated with the sequence and it is often used to describe some constructions in Nature - the patterns of seeds in a sunflower, the spiral of waves, even the breeding of rabbits. My eighth grade math teacher loved the sequence and pointed out that the ratio of adjacent terms gets closer to the golden ratio as the series progresses. For him the golden ratio (φ) was deeply linked to beauty in nature and art. I remember writing out the first twenty or so terms and taking the ratio watching it slowly move towards 1.6180349... (it didn't get that close in the first twenty terms).
For a several years I thought the Fibonacci sequence defined φ. Then I saw something and could ask a deeper question. Here's a start. Pick a couple of random numbers - for simplicity make them positive integers. Add them and then proceed using the rule Xn = Xn-1 + Xn-2 .. related to the Fibonacci sequence, but you are providing your own seed with the random numbers. So if I picked 1435 and 3956 the third term would be 5391, the forth would be 9347 and so on. Do this at least a dozen times and then take the ratio of the last term and that immediately preceding it. Cute, eh? If you program you may want to make a simple program that shows the convergence.
Now I want to do a bit of math, but the tools to write equations that will render on the range of browsers are awful - so I'll get out a pen and sheet of paper. Nothing sophisticated, so don't worry.
Any two integers used as a seed will produce a sequence that converges to the golden ratio. The Fibonacci sequence is an interesting subset of an infinite number of such sequences. Math rules. Just play and questions can lead to insight. Some of these are seen in Nature, but most of the Fibonacci relations are fairly weak. Oh well...
For fun consider Friedman numbers ... numbers that can be written as a mathematical expression using the digits of the number and simple arithemetical opertions like brackets, +, -, *, / and ^.
25 = 52
121 = 112
125 = 5(2+1)
125 = 21*6
many many more. Some preserve the order of the digits and are called nice Friedman nub:
127 = -1 +27
6455 = (64 -5)*5
repeating digits are fun
99999999 = (9 + 9/9)9 - 9/9 - 9/9
11111111111 = ((11 - 1)11 - 1*1)/(11 - 1 - 1)
Other than simple 4 digit and smaller numbers I have found very fewon my own, but they have a certain cool beauty. Check this one out - not only does it have one of all of the digits from 1 to 9, but it is nice.
268435179 = -268 + 4(3*5 -1^7) - 9 (note my text editor doesn't allow double superscripting, so I use a power sign)
1 If you haven't read or listened to Douglas Adams' wonderful The Hitchhiker's Guide to the Galaxy just drop whatever you're doing, get a copy and enjoy. My favorite version is the BBC Radiophonic play studio recording - your library may have it or you can check the normal places - ahem. I can't listen to 'The Journey of the Sorcerer' from the Eagle's One of these Nights album without thinking about it.
The quick summary of the number is a group of hyper-intelligent beings build Deep Thought, a special computer designed to come up with the answer to the ultimate question of life, the Universe, and everything... Deep Thought cranks away for about 7.5 million years and gives the answer --- 42. Of course the hyper-intelligent beings didn't know the question. Getting at that required building the Earth...
2 I'm keeping the convention you probably learned. In modern usage the sequence starts with 0, so F0 = 0 and F1 = 1.
I had to have some corn soup. This is a modification of an old recipe I've been using - you can literally make it a heart stopping dish with heavy cream, but the recent version omitted it and was just fine. It would be better with fresh sweet corn, but this is December and I used a good canned variety. You might want to think about a garnish if you are serving this. Roasted corn with cayenne, basil, ... any number of garnishes would work.
° olive oil (about 2 tbl)
° 1 bunch leeks, sliced thin
° 2 carrots, chopped
° 3 cups of corn
° 4 cups vegetable broth (homemade is best - unsalted is always the way to go)
° 1/2 cup white wine
° 1/2 tsp cayenne papper
° 1 tsp cumin
° 1 bay leaf
° kosher salt
° freshly ground black pepper
° 3/4 cup heavy cream if you dare
° heat a glug (maybe 2 tbl) olive oil over medium heat. Add the leeks and carrots and sauté until tender. Add corn and continue to cook until tender/
° Add the wine and bring to a simmer. Cover and cook until the liquid is almost completely reduced (8 minutes for me) Add the broth and bring back to a simmer.
° Season with the bay leaf, cumin, cayenne, salt and pepper. Simmer for at least 30 minutes to allow a flavor to develop.
° Remove the bay leaf and purée in a blender or (better) with an immersion blender.
° Check the seasoning. If going for a rich soup add the heavy cream here and return to simmer.
° Serve and possibly garnish