Monthly Archives: August 2013

Performativizing Papyrocentricity #13

by in Book Reviews, Notices and tagged , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

Brown Study — William in Trouble, Richmal Crompton (1927)

Bleeding Black — Watch You Bleed: The Saga of Guns n’ Roses, Stephen Davis (Michael Joseph 2008)

Toxic Tails — In the Seventies: Adventures in the Counterculture, Barry Miles (Serpent’s Tail 2011)

Leaf Brief — What a Plant Knows: A Field Guide to the Senses of Your Garden — and Beyond, Daniel Chamovitz (Oneworld 2012) (posted @ Overlord of the Über-Feral)

Electrify Your Eyes — The Spark of Life: Electricity in the Human Body, Frances Ashcroft (Penguin 2013) (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

Narcissarithmetic

by in Base Behavior, Digit-sums, Mathematics, Powers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Why is 438,579,088 a beautiful number? Simple: it may seem entirely arbitrary, but it’s actually self-empowered:

438,579,088 = 4^4 + 3^3 + 8^8 + 5^5 + 7^7 + 9^9 + 0^0 + 8^8 + 8^8 = 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 (usually 0^0 = 1, but the rule is slightly varied here)

438,579,088 is so beautiful, in fact, that it’s in love with itself as a narcissistic number, or number that can be generated by manipulation of its own digits. 89 = 8^1 + 9^2 = 8 + 81 and 135 = 1^1 + 3^2 + 5^3 = 1 + 9 + 125 are different kinds of narcissistic number. 3435 is self-empowered again:

3435 = 3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125

But that’s your lot: there are no more numbers in base-10 that are equal to the sum of their self-empowered digits (apart from the trivial 0 and 1). To prove this, start by considering that there is a limit to the size of a self-empowered number. 9^9 is 387,420,489, which is nine digits long. The function autopower(999,999,999) = 387,420,489 x 9 = 3,486,784,401, which is ten digits long. But autopower(999,999,999,999) = 387,420,489 x 12 = 4,649,045,868, also ten digits long.

The Metamorphosis of Narcissus by Salvador Dalí

Salvador Dalí, La Metamorfosis de Narciso (1937)

So you don’t need to check numbers above a certain size. There still seem a lot of numbers to check: 438,579,088 is a long way above 3435. However, the search is easy to shorten if you consider that checking 3-3-4-5 is equivalent to checking 3-4-3-5, just as checking 034,578,889 is equivalent to checking 438,579,088. If you self-empower a number and the result has the same digits as the original number, you’ve found what you’re looking for. The order of digits in the original number doesn’t matter, because the result has automatically sorted them for you. The function autopower(3345) produces 3435, therefore 3435 must be self-empowered.

So the rule is simple: Check only the numbers in which any digit is greater than or equal to all digits to its left. In other words, you check 12 and skip 21, check 34 and skip 43, check 567 and skip 576, 657, 675, 756 and 765. That reduces the search-time considerably: discarding numbers is computationally simpler than self-empowering them. It’s also computationally simple to vary the base in which you’re searching. Base-10 produces only two self-empowered numbers, but its neighbours base-9 and base-11 are much more fertile:

30 = 3^3 + 0^0 = 30 + 0 (b=9)
27 = 27 + 0 (b=10)

31 = 3^3 + 1^1 = 30 + 1 (b=9)
28 = 27 + 1 (b=10)

156262 = 1^1 + 5^5 + 6^6 + 2^2 + 6^6 + 2^2 = 1 + 4252 + 71000 + 4 + 71000 + 4 (b=9)
96446 = 1 + 3125 + 46656 + 4 + 46656 + 4 (b=10)

1647063 = 1^1 + 6^6 + 4^4 + 7^7 + 0^0 + 6^6 + 3^3 = 1 + 71000 + 314 + 1484617 + 0 + 71000 + 30 (b=9)
917139 = 1 + 46656 + 256 + 823543 + 0 + 46656 + 27 (b=10)

1656547 = 1^1 + 6^6 + 5^5 + 6^6 + 5^5 + 4^4 + 7^7 = 1 + 71000 + 4252 + 71000 + 4252 + 314 + 1484617 (b=9)
923362 = 1 + 46656 + 3125 + 46656 + 3125 + 256 + 823543 (b=10)

34664084 = 3^3 + 4^4 + 6^6 + 6^6 + 4^4 + 0^0 + 8^8 + 4^4 = 30 + 314 + 71000 + 71000 + 314 + 0 + 34511011 + 314 (b=9)
16871323 = 27 + 256 + 46656 + 46656 + 256 + 0 + 16777216 + 256 (b=10)

66500 = 6^6 + 6^6 + 5^5 + 0^0 + 0^0 = 32065 + 32065 + 2391 + 0 + 0 (b=11)
96437 = 46656 + 46656 + 3125 + 0 + 0 (b=10)

66501 = 6^6 + 6^6 + 5^5 + 0^0 + 1^1 = 32065 + 32065 + 2391 + 0 + 1 (b=11)
96438 = 46656 + 46656 + 3125 + 0 + 1 (b=10)

517503 = 5^5 + 1^1 + 7^7 + 5^5 + 0^0 + 3^3 = 2391 + 1 + 512816 + 2391 + 0 + 25 (b=11)
829821 = 3125 + 1 + 823543 + 3125 + 0 + 27 (b=10)

18453278 = 1^1 + 8^8 + 4^4 + 5^5 + 3^3 + 2^2 + 7^7 + 8^8 = 1 + 9519A75 + 213 + 2391 + 25 + 4 + 512816 + 9519A75 (b=11)
34381388 = 1 + 16777216 + 256 + 3125 + 27 + 4 + 823543 + 16777216 (b=10)

18453487 = 1^1 + 8^8 + 4^4 + 5^5 + 3^3 + 4^4 + 8^8 + 7^7 = 1 + 9519A75 + 213 + 2391 + 25 + 213 + 9519A75 + 512816 (b=11)
34381640 = 1 + 16777216 + 256 + 3125 + 27 + 256 + 16777216 + 823543 (b=10)

It’s easy to extend the concept of self-empowered narcisso-numbers. The prime 71 = 131 in base-7 and the prime 83 = 146 in base-7. If 131[b=7] is empowered to the digits of 146[b=7], you get 146[b=7]; and if 146[b=7] is empowered to the digits of 131[b=7], you get 131[b=7], like this:

71 = 131[b=7] → 1^1 + 3^4 + 1^6 = 1 + 81 + 1 = 83 = 146[b=7]

83 = 146[b=7] → 1^1 + 4^3 + 6^1 = 1 + 64 + 6 = 71 = 131[b=7]

But it’s not easy to find more examples. Are there other-empowering pairs like that in base-10? I don’t know.

Leaf Brief

by in Biology, Book Reviews, Botany, Flowers, Science and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of What A Plant Knows by Daniel ChamovitzWhat a Plant Knows: A Field Guide to the Senses of Your Garden – and Beyond, Daniel Chamovitz (Oneworld 2012)

This is a brief but burgeoning book, covering a lot of science and a lot of scientific history. Plants stay in one place and don’t seem to suffer pain or discomfort, so they’re good experimental subjects, particularly for introverts. That’s why Charles Darwin devoted even more time to plants than he did to worms and barnacles. Chamovitz describes Darwin’s ingenious experiments and the even more ingenious experiments of the researchers that followed him. Over millions of years the world has set problems of survival for plants; in solving these problems, plants have set puzzles for scientists. How do plants know when to flower and prepare for winter? How do they resist attacks by insects? Or prey on insects? Or invite visits from pollinators? And how do they communicate with each other? The answers aren’t just chemical: they’re electrical too, as research on the world’s most famous carnivorous plant has proved:

Alexander Volkov and his colleagues at Oakwood University in Alabama first demonstrated that it is indeed electricity that causes the Venus flytrap to close. To test the model, they rigged up very fine electrodes and applied an electrical current to the open lobes of the trap. This made the trap close without any direct touch to its trigger hairs … (ch. 6, “What A Plant Remembers”, pp. 147-8)

Acoustics is also at work in the plant kingdom:

In a process known as buzz pollination, bumblebees stimulate a flower to release its pollen by rapidly vibrating their wing muscles without actually flapping their wings, leading to a high-frequency vibration. … In a similar vein, Roman Zweifel and Fabienne Zeugin from the University of Bern in Switzerland have reported ultrasonic vibrations emanating from pine and oak trees during a drought. These vibrations result from changes in the water content of the water-transporting xylem vessels. While these sounds are passive results of physical forces (in the same way that a rock crashing off a cliff makes a noise), perhaps these ultrasonic vibrations are used as a signal by other trees to prepare for dry conditions. (ch. 4, “What A Plant Hears”, pg. 107-8)

All of this is mathematical: a plant is a mechanism that processes not just sun, water and carbon-dioxide, but information from its environment too. But then sun, water and CO2 are all part of that information: sunlight signals plants as well as sustaining them. Its strength and duration are cues for the seasons and time of the day. So is its colour:

By the time John F. Kennedy was elected president, Warren L. Butler and his colleagues had demonstrated that a single photoreceptor in plants was responsible for both the red and far-red effects. They called this receptor “phytochrome”, meaning “plant colour”. In its simplest model, phytochrome is a light-activated switch. Red light activates phytochrome, turning it into a form primed to receive far-red light. Far-red light inactivates phytochrome, turning it into a form primed to receive red light. Ecologically, this makes a lot of sense. In nature, the last light a plant sees at the end of the day is far-red, and this signifies to the plant that it should “turn-off”. In the morning it sees red light and it wakes up. In this way a plant measures how long ago it last saw red light and adjusts its growth accordingly. (ch. 1, “What A Plant Sees”, pg. 21-2)

There’s an obvious analogy with a computer automatically turning itself off and on, which would make phytochrome and its associated chemicals a kind of hardware created by the software of the genes. Plants share some of that software with human beings: in one fascinating section, Chamovitz discusses the links between healthy plants and sick people:

The arabidopsis [A. thaliana, mustard plant] genome contains BRCA, CFTR, and several hundred other genes associated with human disease or impairment because they are essential for basic cellular biology. These important genes had already evolved 1.5 billion years ago in the single-celled organism that was the common evolutionary ancestor to both plants and animals. (ch. 4, “What A Plant Hears”, pg. 105)

What a Plant Knows stimulates human minds as it discusses plant senses. It’s one of the best briefest, or briefest best, books on science I’ve ever read, packing a lot of history and scientific information into six chapters. Plants don’t move much, but they’re a very lively topic and botany is a good way to understand and appreciate biology and scientific research better.

Electrify Your Eyes

by in Biology, Book Reviews, Chemistry, Consciousness, Science and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of The Spark of Life by Frances AshcroftThe Spark of Life: Electricity in the Human Body, Frances Ashcroft (Penguin 2013)

“Electricity in the Human Body” is the subtitle of this book. Make that the goat, frog, eel, shark, torpedo-ray, snake, platypus, spiny anteater, sooty shearwater and fruit-fly body too. And if Venus flytraps, maize and algae have bodies, throw them in next. Frances Ashcroft gives you a bargeload of buzz for your buck, a shedload of shock for your shekel: The Spark of Life describes the use of electricity by many different forms of life. But it discusses death a lot too, from lightning-strikes and electric chairs to heart-attacks and toxicology. Poisons can be a cheap and highly effective way of interfering with the electro-chemistry of the body:

The importance of sodium and potassium channels in generating the nerve impulse is demonstrated by the fact that a vast array of poisons from spiders, shellfish, sea anemones, frogs, snakes, scorpions and many other exotic creatures interact with these channels and thereby modify the function of nerve and muscle. … The tetrodotoxin contained in the liver and other tissues of this fish [the fugu or puffer-fish, Takifugu spp., Lagocephalus spp., etc] is a potent blocker of the sodium channels found in your nerves and skeletal muscles. It causes numbness and tingling of the lips and mouth within as little as thirty minutes … This sensation of “pins and needles” spreads rapidly to the face and neck, moves onto the fingers and toes, and is then followed by gradual paralysis of the skeletal muscles … Ultimately the respiratory muscles are paralysed, which can be fatal. The heart is not affected, as it has a different kind of sodium channel that is far less sensitive to tetrodotoxin. The toxin is also unable to cross the blood-brain barrier so that, rather horrifyingly, although unable to move and near death, the patient remains conscious. (ch. 3, “Acting on Impulse”, pp. 69-70)

In short, fugu-poisoning is the opposite of electrocution: it’s the absence rather than the excess of electricity that kills its victims. Those “channels” are a reminder that electro-chemistry could also be called electro-mechanics: unlike an electricity-filled computer, an electricity-filled body has moving parts – and in more ways than one. Our muscles move because ions move in and out of our cells. This means that a body has to be wet inside, not dry like a computer, but it’s easy to imagine a human brain controlling a robotic body. But would a brain still be conscious if it became metal-and-plastic too? Perhaps a brain has to be both soggy and sparky to be conscious.

The electrical nature of the brain certainly seems important, though that may be a superstitious conclusion. Electricity is a mysterious phenomenon and so is consciousness, so they seem to go together well. Ashcroft writes a lot about the sense-organs and the data they supply to the brain, but like all scientists she cannot explain how those data are turned into conscious experience as the maths-engine of the brain applies its neuro-functions and neuro-algorithms. However, she does suggest ways in which our consciousness might be expanded in future. Humans have colour vision, based on the three types of cone-cells in our eyes:

Most mammals, such as cats and dogs, have only two types of cone photopigment and so see only a limited range of colour … Other animals live in a world entirely without colour. But humans should not be too complacent, for we are far from having the best colour vision in the animal world and lag far behind the mantis shrimp, which enjoys ten or more different visual pigments. Even tropical fish possess four or five types of cones. (ch. 9, “The Doors of Perception”, pg. 199)

Bio-engineering may one day sharpen and extend all our senses, from sight and hearing to touch, taste and smell. It may also give us new senses, like the ability to form sound-pictures like bats and detect infra-red like pit-vipers. And why not X-rays and radio-waves too? It’s an exciting prospect, but in a sense it won’t be anything new: our new senses, like our old ones, will depend on nerve-impulses and the way they’re mashed and mathed in that handful of “electrified clay” known as the brain.

“Electrified clay” is Shelley’s phrase: like his wife Mary, he was fascinated by the early electric experiments of the Italian scientists Luigi Galvani and Alessandro Volta. Mary turned her fascination into a book called Frankenstein (1818) and her invention is part of the scientific history in this book. The story of bio-electricity is still going strong: there are electric mysteries in all kinds of bodies waiting to be solved. Maybe consciousness is one of them. And if science proves unable to crack consciousness, it’s certainly able to expand it. Reading this book is one way to experience the mind-expanding powers of science, but seeing like a mantis shrimp would be good too.

In Perms Of

by in Base Behavior, Mathematics, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

13 is a prime number, divisible only by itself and 1. Perm 13 and you get 31, which is also a prime number. The same is true of 17, 37 and 79. There are only two possible permutations – 2 x 1 – of a two-digit number, so base-10 is terminally permal for two-digit primes:

13, 31
17, 71
37, 73
79, 97

What about three-digit primes? Now there are six possible permutations: 3 x 2 x 1. But base-10 is not terminally permal for three-digit primes. This is the best it does:

149, 419, 491, 941
179, 197, 719, 971
379, 397, 739, 937

Fortunately, we aren’t restricted to base-10. Take a step up and you’ll find that base-11 is terminally permal for three-digit primes (139 in base-11 = 1 x 11^2 + 3 x 11 + 9 = 163 in base-10):

139, 193, 319, 391, 913, 931 (6 primes) (base=11)

163, 223, 383, 463, 1103, 1123 (base=10)

Four-digit primes have twenty-four possible permutations – 4 x 3 x 2 x 1 – and base-10 again falls short:

1237, 1327, 1723, 2137, 2371, 
2713, 2731, 3217, 3271, 7213,
7321 (11 primes)

1279, 1297, 2179, 2719, 2791,
2917, 2971, 7129, 7219, 9127,
9721

For four-digit primes, the most permal base I’ve discovered so far is base-13 (where B represents [11]):

134B, 13B4, 14B3, 1B34, 1B43,
314B, 31B4, 34B1, 3B14, 413B,
41B3, 431B, 43B1, 4B13, 4B31,
B134, B143, B314, B413 (19 primes) (base=13)

2767, 2851, 3019, 4099, 4111,
6823, 6907, 7411, 8467, 9007,
9103, 9319, 9439, 10663, 10687,
24379, 24391, 24691, 24859 (base=10)

Is there a base in which all permutations of some four-digit number are prime? I think so, but I haven’t found it yet. Is there always some base, b, in which all permutations of some d-digit number are prime? Is there an infinity of bases in which all permutations of some d-digit number are prime? Easy to ask, difficult to answer. For me, anyway.

Ass You Like It

by in Über-Transgression..., English usage, Free Will, Guardianistas, Philosophy, Politics, Post-Transgressive Fiction and tagged , , , , , , , , , , , , , , , , , , , , | 8 Comments

This is a guest post by Norman Foreman, B.A.

Mediaeval Catholic philosophers wrote about both praying and braying. The braying came from Buridan’s ass, a thought-experiment about choice and free will. Imagine a hungry ass set between two piles of hay that are identical in every way: size, shape, colour, tastiness and so on. Some philosophers argued that, if it had no reason to prefer one pile of hay to the other, the ass would be unable to choose and would therefore starve to death.

I don’t agree: inter alia, nervous systems don’t work symmetrically and we don’t experience objects as fully identical when they’re in different parts of our visual field. However, in a literary sense, I understand what it feels like to be Buridan’s ass. To assify myself, I start by imagining this:

• I’m offered £1000 to read a book by the transgressive author Will Self.

Would I accept? Yes. It would be distasteful, but I’d do it for £1000. Self’s writing is so bad that I might give the money back rather than finish the book, but I’d have a go. Now change the situation:

• I’m offered £1000 to read a book by the transgressive author Stewart Home.

Would I still accept? Yes. Again, it would be distasteful, but I’d do it for the money. Or I’d try, at least. The next step turns me into Buridan’s ass. I imagine this:

• I’m offered £1000 to read a book by either Will Self or Stewart Home (not both). And I have to make the choice for myself.

Now I’m on the horns of a dilemma. I would want the £1000, but I can’t decide which transgressive author I’d rather NOT read. Home is a downmarket version of Self, Self is an upmarket version of Home. It’s Self-as-chav vs Home-as-Oxbridge-grad. And/or vice versâ. They’re both keyly committed components of the Guardianista community, with all that that implies in terms of issues around bad English, mixed metaphors and “in terms of”. I’m happy to say I’ve never read a book by either of them. So if I were offered £1000 to do so and had to choose either Self or Home, I couldn’t do it. Not unassisted. I’d have to toss a coin. Best of three. Or best of five dot dot dot

Previously pre-posted (please peruse):

• Titus Graun
• Ex-term-in-ate!
• Reds under the Thread

Performativizing Papyrocentricity #12

by in Book Reviews, Surfing, The Sea and tagged , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

Hawt’ in the Act – Whatshisname: The Life and Death of Charles Hawtrey, Wes Butters (Tomahawk Press 2010)

Lez Redd – The Trials and Triumphs of Les Dawson, Louis Barfe (Atlantic Books 2012)

Fetch and Carry – The Surfrider, compiled by Jack Pollard (K.G. Murray 1963)

Bri’ on the Sky – Wonders of the Solar System, Professor Brian Cox and Andrew Cohen (Collins 2010) (posted @ Overlord of the Über-Feral)

Playing on the Nerves – In A Glass Darkly, Sheridan Le Fanu (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

Bri’ on the Sky

by in Astronomy, Book Reviews, Politics, Science and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 5 Comments

Front cover of Wonders of the Solar System by Brian Cox and Andrew Cohen

Bri’ Eyes the Sky

Wonders of the Solar System, Professor Brian Cox and Andrew Cohen (Collins 2010)

One of the most powerful images in this book is also one of the most understated. It’s an artist’s impression of a dim star seen over the curve of a dwarf-planet called Sedna. The star is a G-type called Sol. We on Earth know it better as the sun. Sedna is a satellite of the sun too, but it’s much, much further out than we are. It takes 12,000 years to complete a single orbit and its surface is a biophobic -240°C. It’s so distant that sunrise is star-rise and it wasn’t discovered until 2003. But the sun’s gravity still keeps it in place: one of the weakest forces in nature is one of the most influential. That’s one important message in an understated, crypto-Lovecraftian image.

Sedna has been there, creeping around its dim mother-star, since long before man evolved. It will still be there long after man disappears, voluntarily or otherwise. This frozen dwarf is a good symbol of the vastness of the universe and its apparent indifference to life. We don’t seem to interest the universe at all, but the universe certainly interests us. Wonders of the Solar System is a good introduction to our tiny corner of it, describing some fundamentals of astronomy with the help of spectacular photographs and well-designed illustrations. You can learn how fusion powers the sun, how Mars lost its atmosphere and how there might be life beneath the frozen surface of Jupiter’s satellite Europa. The text is simple, but not simplistic, though I think the big name on the cover did little of the writing: this book is probably much more Cohen than Cox. Either way, I enjoyed reading the words and not just looking at the pictures, all the way from star-dim Sedna (pp. 26-7) to “Scars on Mars” (pp. 220-1) by way of “The most violent place in the solar system” (pp. 198-9), a.k.a. Jupiter’s gravity-flexed, volcano-pocked satellite Io.

Pockmarked moon -- the Galilean satellite Io

Pockmarked moon — the Galilean satellite Io

Everything described out there is linked to something down here, because that’s how it was done in the television series. Linking the sky with the earth allowed the BBC to film the genial and photogenic physicist Brian Cox in various exotic settings: Hawaii, India, East Africa, Iceland and so on. I’ve not seen any of Cox’s TV-work, but he seems an effective popularizer of science. And the pretty-boy shots here add anthropology to the astronomy. What is the scientific point of Cox striding away in an artistic blur over the Sahara desert (pg. 103), staring soulfully into the distance near the Iguaçu Falls on the Brazilian-Argentine border (pg. 37) or gazing down into the Grand Canyon, hips slung, hands in pockets (pg. 163)? There isn’t a scientific point: the photos are there for his fans, particularly his female ones. He’s a sci-celeb, a geek with chic, and we’re supposed to see the sky through Bri’s eyes.

But he’s also a liberal working for the Bolshevik Broadcasting Corporation, so he’ll be happy with the prominent photo early on: Brian holding protective glasses over the eyes of a dusky-skinned child during a solar eclipse in India. The same simul-scribes’ Wonders of Life (Collins 2013), another book-of-the-BBC-series, opens with a similarly allophilic allophoto: a dusky-skinned Mexican crowned in monarch butterflies. This is narcissistic and patronizing, but the readiness of whites to “Embrace the Other” helps explain science, because science involves looking away from the self, the tribe and the quotidian quest for status and survival. Of course, Cox and Cohen would gasp with horror at the idea of racial differences explaining big things like science and politics. Cox would be sincere in his horror. I’m not so sure about Cohen.

But there are wonders within us as well as without us and though you won’t hear about them on the BBC, the tsunami of HBD, or research into human bio-diversity, is now rolling ashore. It will sweep away almost all of Cox’s and Cohen’s politics, but leave most of their science intact. It isn’t a coincidence that the rings of Saturn were discovered by the Italian Galileo and explained by the Dutchman Huygens and the Italian Cassini, or that the photos of Saturn here were taken by a space-probe launched by white Americans. But the United States has much less money now for space exploration. That’s explained by race too: as the US looks less like its founders, it looks less like a First World nation too. It’s fun to see the world through Bri’s eyes, but he’s careful not to look at everything that’s out there.

Factory Records

by in Base Behavior, Mathematics, Palindromic numbers, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The factors of n are those numbers that divide n without remainder. So the factors of 6 are 1, 2, 3 and 6. If the function s(n) is defined as “the sum of the factors of n, excluding n”, then s(6) = 1 + 2 + 3 = 6. This makes 6 a perfect number: its factors re-create it. 28 is another perfect number. The factors of 28 are 1, 2, 4, 7, 14 and 28, so s(28) = 1 + 2 + 4 + 7 + 14 = 28. Other perfect numbers are 496 and 8128. And they’re perfect in any base.

Amicable numbers are amicable in any base too. The factors of an amicable number sum to a second number whose factors sum to the first number. So s(220) = 284, s(284) = 220. That pair may have been known to Pythagoras (c.570-c.495 BC), but s(1184) = 1210, s(1210) = 1184 was discovered by an Italian schoolboy called Nicolò Paganini in 1866. There are also sociable chains, in which s(n), s(s(n)), s(s(s(n))) create a chain of numbers that leads back to n, like this:

12496 → 14288 → 15472 → 14536 → 14264 → 12496 (c=5)

Or this:

14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (c=28)

Those sociable chains were discovered (and christened) in 1918 by the Belgian mathematician Paul Poulet (1887-1946). Other factor-sum patterns are dependant on the base they’re expressed in. For example, s(333) = 161. So both n and s(n) are palindromes in base-10. Here are more examples — the numbers in brackets are the prime factors of n and s(n):

333 (3^2, 37) → 161 (7, 23)
646 (2, 17, 19) → 434 (2, 7, 31)
656 (2^4, 41) → 646 (2, 17, 19)
979 (11, 89) → 101 (prime)
1001 (7, 11, 13) → 343 (7^3)
3553 (11, 17, 19) → 767 (13, 59)
10801 (7, 1543) → 1551 (3, 11, 47)
11111 (41, 271) → 313 (prime)
18581 (17, 1093) → 1111 (11, 101)
31713 (3, 11, 31^2) → 15951 (3, 13, 409)
34943 (83, 421) → 505 (5, 101)
48484 (2^2, 17, 23, 31) → 48284 (2^2, 12071)
57375 (3^3, 5^3, 17) → 54945 (3^3, 5, 11, 37)
95259 (3, 113, 281) → 33333 (3, 41, 271)
99099 (3^2, 7, 11^2, 13) → 94549 (7, 13, 1039)
158851 (7, 11, 2063) → 39293 (prime)
262262 (2, 7, 11, 13, 131) → 269962 (2, 7, 11, 1753)
569965 (5, 11, 43, 241) → 196691 (11, 17881)
1173711 (3, 7, 11, 5081) → 777777 (3, 7^2, 11, 13, 37)

Note how s(656) = 646 and s(646) = 434. There’s an even longer sequence in base-495:

33 → 55 → 77 → 99 → [17][17] → [19][19] → [21][21] → [43][43] → [45][45] → [111][111] → [193][193] → [195][195] → [477][477] (b=495) (c=13)
1488 (2^4, 3, 31) → 2480 (2^4, 5, 31) → 3472 (2^4, 7, 31) → 4464 (2^4, 3^2, 31) → 8432 (2^4, 17, 31) → 9424 (2^4, 19, 31) → 10416 (2^4, 3, 7, 31) → 21328 (2^4, 31, 43) → 22320 (2^4, 3^2, 5, 31) → 55056 (2^4, 3, 31, 37) → 95728 (2^4, 31, 193) → 96720 (2^4, 3, 5, 13, 31) → 236592 (2^4, 3^2, 31, 53)

I also tried looking for n whose s(n) mirrors n. But they’re hard to find in base-10. The first example is this:

498906 (2, 3^3, 9239) → 609894 (2, 3^2, 31, 1093)

498906 mirrors 609894, because the digits of each run in reverse to the digits of the other. Base-9 does better for mirror-sums, clocking up four in the same range of integers:

42 → 24 (base=9)
38 (2, 19) → 22 (2, 11)
402 → 204 (base=9)
326 (2, 163) → 166 (2, 83)
4002 → 2004 (base=9)
2918 (2, 1459) → 1462 (2, 17, 43)
5544 → 4455 (base=9)
4090 (2, 5, 409) → 3290 (2, 5, 7, 47)

Base-11 does better still, clocking up eight in the same range:

42 → 24 (base=11)
46 (2, 23) → 26 (2, 13)
2927 → 7292 (base=11)
3780 (2^2, 3^3, 5, 7) → 9660 (2^2, 3, 5, 7, 23)
4002 → 2004 (base=11)
5326 (2, 2663) → 2666 (2, 31, 43)
13772 → 27731 (base=11)
19560 (2^3, 3, 5, 163) → 39480 (2^3, 3, 5, 7, 47)
4[10]7[10]9 → 9[10]7[10]4 (base=11)
72840 (2^3, 3, 5, 607) → 146040 (2^3, 3, 5, 1217)
6929[10] → [10]9296 (base=11)
100176 (2^4, 3, 2087) → 158736 (2^4, 3, 3307)
171623 → 326171 (base=11)
265620 (2^2, 3, 5, 19, 233) → 520620 (2^2, 3, 5, 8677)
263702 → 207362 (base=11)
414790 (2, 5, 41479) → 331850 (2, 5^2, 6637)

Note that 42 mirrors its factor-sum in both base-9 and base-11. But s(42) = 24 in infinitely many bases, because when 42 = 2 x prime, s(42) = 1 + 2 + prime. So (prime-1) / 2 will give the base in which 24 = s(42). For example, 2 x 11 = 22 and 22 = 42 in base (11-1) / 2 or base-5. So s(42) = 1 + 2 + 11 = 14 = 2 x 5 + 4 = 24[b=5]. There are infinitely many primes, so infinitely many bases in which s(42) = 24.

Base-10 does better for mirror-sums when s(n) is re-defined to include n itself. So s(69) = 1 + 3 + 23 + 69 = 96. Here are the first examples of all-factor mirror-sums in base-10:

69 (3, 23) → 96 (2^5, 3)
276 (2^2, 3, 23) → 672 (2^5, 3, 7)
639 (3^2, 71) → 936 (2^3, 3^2, 13)
2556 (2^2, 3^2, 71) → 6552 (2^3, 3^2, 7, 13)

In the same range, base-9 now produces one mirror-sum, 13 → 31 = 12 (2^2, 3) → 28 (2^2, 7). Base-11 produces no mirror-sums in the same range. Base behaviour is eccentric, but that’s what makes it interesting.