Suppose you set up an L, i.e. a vertical and horizontal line, representing the x,y coordinates between 0 and 1. Next, find the fractional pairs x = 1/2, 1/3, 2/3, 1/4, 2/4…, y = 1/2, 1/3, 2/3, 1/4, 2/4… and mark the point (x,y). That is, find the point, say, 1/5 of the way along the x-line, then the points 1/5, 2/5, 3/5 and 4/5 along the y-line, marking the points (1/5, 1/5), (1/5, 2/5), (1/5, 3/5), (1/5, 4/5). Then find (2/5, 1/5), (2/5, 2/5), (2/5, 3/5), (2/5, 4/5) and so on. Some interesting patterns appear in what I call a Frac-L (pronounced “frackle”) or Fract-L:
Frac-L for 1/2 to 21/22
Frac-L for 1/2 to 48/49
Frac-L for 1/2 to 75/76
Frac-L for 1/2 to 102/103
Frac-L for 1/2 to 102/103 (animated)
If the (x,y) point is first red, then becomes different colors as it is repeatedly found, you get these patterns:
Frac-L for 1/2 to 48/49 (color)
Frac-L for 1/2 to 75/79 (color)
Frac-L for 1/2 to 102/103 (color) (animated)
Now try polygonal numbers. The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78…, so you’re finding the fractional pairs, say, (1/21, 1/21), (1/21, 3/21, (1/21, 6/21), (1/21, 10/21), (1/21, 15/21), then (3/21, 1/21), (3/21, 3/21, (3/21, 6/21), (3/21, 10/21), (3/21, 15/21), and so on:
Frac-L for triangular fractions
The frac-L for square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100…) is almost identical:
Frac-L for square fractions, e.g. (1/16, 1/16), (1/16, 4/16), (1/16, 9/16)…
So is the frac-L for pentagonal numbers (1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330…):
Frac-L for pentagonal fractions, e.g. (1/35, 5/35), (1/35, 12/35), (1/35,22/35)…
But what about prime numbers (skipping 2)? Here the fractional pairs are, say, (1/17, 1/17), (1/17, 3/17), (1/17, 5/17), (1/17, 7/17), (1/17, 11/17), (1/17, 13/17), then (3/17, 1/17), (3/17, 3/17), (3/17, 5/17), (3/17, 7/17), (3/17, 11/17), (3/17, 13/17), and so on:
Frac-L for 1/3 to 73/79 (prime fractions)
Frac-L for 1/3 to 223/227
Frac-L for 1/3 to 307/331
Frac-L for 1/3 to 307/331 (animated)
Frac-L for 1/3 to 73/79 (color) (prime fractions)
Frac-L for 1/3 to 223/227 (color)
Frac-L for 1/3 to 307/331 (color)
Frac-L for 1/3 to 307/331 (color) (animated)
And finally (for now), a frac-L for Fibonnaci numbers, where the fractional pairs are, say, (1/13, /13), (1/13, 2/13), (1/13, 3/13), (1/13, 5/13), (1/13, 8/13), then (2/13, /13), (2/13, 2/13), (2/13, 3/13), (2/13, 5/13), (2/13, 8/13), and so on:
Frac-L for Fibonacci fractions to 14930352/2178309 = fibonacci(36)/fibonacci(37)
It’s a very simple function that raises a very difficult question. An unanswered question, in fact. Take any whole number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. Repeat until you reach 1. That’s the hailstone function, because the numbers rise and fall like hailstones being formed in a cloud. Here are some examples:
But is this function truly a hailstone function? That is, does every number fall finally to earth and reach 1? So far, for every number tested, the answer has been yes. But do all numbers reach 1? The Collatz conjecture says they do. But no-one can prove it. Or disprove it. All it would take is one number failing to fall to earth. Mathematicians don’t think there is one, but numbers can take a surprising length of time to get to the ground. Here’s 27:
27 takes 111 steps to reach 1. And the 111 made me think of another question. If the function hail(n) returns the number of steps required for n to reach 1, then hail(27) = 111. But what about hail(n) = 666? That is, what is the first number that requires 666 steps to reach 1? I say “first number”, because one very big number is guaranteed to take 666 steps:
Put another way, 666 = hail(2^666), because for any power of 2, hail(2^p) = p. But is there a smaller number, which I’ll call satan, for which hail(satan) = 666? Here’s a tantalizing taster of the task:
Here’s a question I haven’t answered: if satanic numbers are those n satisfying hail(n) = 666, how many satanic numbers are there? We’ve already seen two of them: 666 = hail(2^666) = hail(26597116). But how many more are there? Not infinitely many, because for n > 2^666, hail(n) > 666. In fact, after satan = 26597116, the next three satanic numbers arrive very quickly:
So there are four consecutive satanic numbers. But it isn’t unusual for a run of consecutive numbers to have the same hail(). Here’s a graph of the values of hail(n) for n = 1,2,3… (running left-to-right, down-up, with 1,2,3… in the lower lefthand corner). When n is divisible by 10, hail(n) is represented in red; when n is odd and divisible by 5, hail(n) is green. Note how many runs of identical hail(n) there are:
Graph for hail(n)
Here are successive records for runs of identical hail(n):
At least, I think it’s a fractal. I came across it when I was counting the ways in which the integers can be the sum of distinct Fibonacci numbers. Here for reference is the Fibonacci sequence, the beautiful and endlessly fertile sequence that’s seeded with “1, 1” and continued by summing the two previous numbers:
I also noticed a pattern relating to the maximum count reached in the numbers between the 1s. Suppose the function max(fib(i)-1..fib(i+1)-1) returns the highest count of ways to represent the numbers from fib(i)-1 to fib(i+1)-1. Notice how max() increases:
The pattern is described like this at the Online Encyclopedia of Integer Sequences:
a(n) = 1 if and only if n+1 is a Fibonacci number. The length of such a quasi-period (from Fib(i)-1 to Fib(i+1)-1, inclusive) is a Fibonacci number + 1. The maximum value of a(n) within each subsequent quasi-period increases by a Fibonacci number. For example, from n = 143 to n = 232, the maximum is 13. From 232 to 376, the maximum is 16, an increase of 3. From 376 to 609, 21, an increase of 5. From 609 to 986, 26, increasing by 5 again. Each two subsequent maxima seem to increase by the same increment, the next Fibonacci number. – Kerry Mitchell, Nov 14 2009
The maxima of the quasi-periods are in A096748. – Max Barrentine, Sep 13 2015 — See commentary for A000119 at OEIS
I can’t remember where I came across this clever little puzzle and what precise form it took, but here’s my version of it:
A famously eccentric inventor and recreational mathematician has invited you to tour the factory where his company manufactures locks, keys, safes, cash-boxes and so on. At the end of the tour he brings you to a conference room, pours you a glass of wine, and invites you to test your wits against a puzzle. He points out that a hundred numbered boxes have been set out on two long tables in the room. You sip your wine as you listen to him explain that each box is locked and contains a slip of paper bearing a number between 0 and 9. If you accept the challenge, the inventor will order a hundred workers to walk in turn past the boxes, using a master-key to unlock or lock the boxes like this:
The first worker will use the key on every box (boxes #1,2,3…), the second worker will use the key on every second box (boxes #2,4,6…), the third worker the key on every third box (boxes #3,6,9…), and so on.
Now, you can’t tell by simply looking at a box whether it’s unlocked or not, but it’s obvious that the first box will be unlocked when all that is over. Box #1 is originally locked and the master-key will be used on it just once. But how many other boxes will be unlocked? If you can choose nothing but the unlocked boxes, you get to keep the contents. Otherwise you get nothing. That is, if you choose one or more locked boxes, you get nothing.
And what good are the contents of the unlocked boxes? Well, if you take the numbered slips of paper they contain in order, they will give you the combination of a locked safe the inventor now points out in the wall behind you. The safe contains the antidote for the deadly but slow-acting poison he secretly slipped into the wine you have been sipping as you listened to him explain the details of the puzzle. So you have to choose all and only the unlocked boxes to save your life. Can you do it?
Solution
I’m sure there’s a simpler explanation of which boxes will be unlocked, but here’s my complicated one:
Whether box #n is locked or unlocked in the end depends on how many divisors the number n has. If it has an even number of divisors, it will be locked; if it has an odd number of divisors, it will be unlocked. Take box #12. The number 12 has six divisors: 1, 2, 3, 4, 6 and 12. So workers #1, #3 and #6 will unlock it with the master-key, but workers #2, #4 and #12 will lock it again. Worker #12 will be the final worker to use the master-key on the box, so it will be locked.
Now take box #16. The number #16 has five divisors: 1, 2, 4, 8 and 16. So workers #1, #4 and #16 will unlock the box with the master-key, while workers #2 and #8 will lock it. Worker #16 will be the final worker to use the master-key on the box, so it will be unlocked.
In other words, the puzzle reduces to this: Which numbers from 1 to 100 have an odd number of divisors? To work out the number of divisors n has, you add 1 to the exponent of each of its prime factors and multiply the results. For example, 24 has eight divisors thus:
• 36 = 2^2 * 3^2 → (2+1) * (2+1) = 3 * 3 = 9, so 36 has nine divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36
36 demonstrates that a number has to have only even exponents on its prime factors to have an odd number of divisors (the only number without prime factors is 1, which has one divisor, namely itself). Numbers with only even exponents on their prime factors are square numbers:
• 4 = 2^2 → (2+1) = 3, so 4 has three divisors: 1, 2, 4
• 9 = 3^2 → (2+1) = 3, so 9 has three divisors: 1, 3, 9
• 16 = 2^4 → (4+1) = 5, so 16 has five divisors: 1, 2, 4, 8, 16
• 25 = 5^2 → (2+1) = 3, so 25 has divisors: 1, 5, 25
• 36 = 2^2 * 3^2 → (2+1) * (2+1) = 3 * 3 = 9, so 36 has nine divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36
• 49 = 7^2 → (2+1) = 3, so 49 has three divisors: 1, 7, 49
• 64 = 2^6 → (6+1) = 7, so 64 has seven divisors: 1, 2, 4, 8, 16, 32, 64
• 81 = 3^4 → (4+1) = 5, so 81 has five divisors: 1, 3, 9, 27, 81
• 100 = 2^2 * 5^2 → (2+1) * (2+1) = 3 * 3 = 9, so 100 has nine divisors: 1, 2, 4, 5, 10, 20, 25, 50, 100
So if you choose boxes #1, #4, #9, #16, #25, #36, #49, #64, #81 and #100, you’ll get the combination for the safe and save your life.
Appendix
Here’s the full description of what happens to the boxes:
• box #1 is unlocked by worker #1 and locked by no-one, therefore it’s unlocked
• box #2 is unlocked by worker #1 and locked by worker #2, therefore it’s locked
• box #3 is unlocked by worker #1 and locked by worker #3, therefore it’s locked
• box #4 is unlocked by workers #1 and #4, and locked by worker #2, therefore it’s unlocked
• box #5 is unlocked by worker #1 and locked by worker #5, therefore it’s locked
• box #6 is unlocked by workers #1 and #3, and locked by workers #2 and #6, therefore it’s locked
• box #7 is unlocked by worker #1 and locked by worker #7, therefore it’s locked
• box #8 is unlocked by workers #1 and #4, and locked by workers #2 and #8, therefore it’s locked
• box #9 is unlocked by workers #1 and #9, and locked by worker #3, therefore it’s unlocked
• box #10 is unlocked by workers #1 and #5, and locked by workers #2 and #10, therefore it’s locked
• box #11 is unlocked by worker #1 and locked by worker #11, therefore it’s locked
• box #12 is unlocked by workers #1, #3 and #6, and locked by workers #2, #4 and #12, therefore it’s locked
• box #13 is unlocked by worker #1 and locked by worker #13, therefore it’s locked
• box #14 is unlocked by workers #1 and #7, and locked by workers #2 and #14, therefore it’s locked
• box #15 is unlocked by workers #1 and #5, and locked by workers #3 and #15, therefore it’s locked
• box #16 is unlocked by workers #1, #4 and #16, and locked by workers #2 and #8, therefore it’s unlocked
• box #17 is unlocked by worker #1 and locked by worker #17, therefore it’s locked
• box #18 is unlocked by workers #1, #3 and #9, and locked by workers #2, #6 and #18, therefore it’s locked
• box #19 is unlocked by worker #1 and locked by worker #19, therefore it’s locked
• box #20 is unlocked by workers #1, #4 and #10, and locked by workers #2, #5 and #20, therefore it’s locked
• box #21 is unlocked by workers #1 and #7, and locked by workers #3 and #21, therefore it’s locked
• box #22 is unlocked by workers #1 and #11, and locked by workers #2 and #22, therefore it’s locked
• box #23 is unlocked by worker #1 and locked by worker #23, therefore it’s locked
• box #24 is unlocked by workers #1, #3, #6 and #12, and locked by workers #2, #4, #8 and #24, therefore it’s locked
• box #25 is unlocked by workers #1 and #25, and locked by worker #5, therefore it’s unlocked
• box #26 is unlocked by workers #1 and #13, and locked by workers #2 and #26, therefore it’s locked
• box #27 is unlocked by workers #1 and #9, and locked by workers #3 and #27, therefore it’s locked
• box #28 is unlocked by workers #1, #4 and #14, and locked by workers #2, #7 and #28, therefore it’s locked
• box #29 is unlocked by worker #1 and locked by worker #29, therefore it’s locked
• box #30 is unlocked by workers #1, #3, #6 and #15, and locked by workers #2, #5, #10 and #30, therefore it’s locked
• box #31 is unlocked by worker #1 and locked by worker #31, therefore it’s locked
• box #32 is unlocked by workers #1, #4 and #16, and locked by workers #2, #8 and #32, therefore it’s locked
• box #33 is unlocked by workers #1 and #11, and locked by workers #3 and #33, therefore it’s locked
• box #34 is unlocked by workers #1 and #17, and locked by workers #2 and #34, therefore it’s locked
• box #35 is unlocked by workers #1 and #7, and locked by workers #5 and #35, therefore it’s locked
• box #36 is unlocked by workers #1, #3, #6, #12 and #36, and locked by workers #2, #4, #9 and #18, therefore it’s unlocked
• box #37 is unlocked by worker #1 and locked by worker #37, therefore it’s locked
• box #38 is unlocked by workers #1 and #19, and locked by workers #2 and #38, therefore it’s locked
• box #39 is unlocked by workers #1 and #13, and locked by workers #3 and #39, therefore it’s locked
• box #40 is unlocked by workers #1, #4, #8 and #20, and locked by workers #2, #5, #10 and #40, therefore it’s locked
• box #41 is unlocked by worker #1 and locked by worker #41, therefore it’s locked
• box #42 is unlocked by workers #1, #3, #7 and #21, and locked by workers #2, #6, #14 and #42, therefore it’s locked
• box #43 is unlocked by worker #1 and locked by worker #43, therefore it’s locked
• box #44 is unlocked by workers #1, #4 and #22, and locked by workers #2, #11 and #44, therefore it’s locked
• box #45 is unlocked by workers #1, #5 and #15, and locked by workers #3, #9 and #45, therefore it’s locked
• box #46 is unlocked by workers #1 and #23, and locked by workers #2 and #46, therefore it’s locked
• box #47 is unlocked by worker #1 and locked by worker #47, therefore it’s locked
• box #48 is unlocked by workers #1, #3, #6, #12 and #24, and locked by workers #2, #4, #8, #16 and #48, therefore it’s locked
• box #49 is unlocked by workers #1 and #49, and locked by worker #7, therefore it’s unlocked
• box #50 is unlocked by workers #1, #5 and #25, and locked by workers #2, #10 and #50, therefore it’s locked
• box #51 is unlocked by workers #1 and #17, and locked by workers #3 and #51, therefore it’s locked
• box #52 is unlocked by workers #1, #4 and #26, and locked by workers #2, #13 and #52, therefore it’s locked
• box #53 is unlocked by worker #1 and locked by worker #53, therefore it’s locked
• box #54 is unlocked by workers #1, #3, #9 and #27, and locked by workers #2, #6, #18 and #54, therefore it’s locked
• box #55 is unlocked by workers #1 and #11, and locked by workers #5 and #55, therefore it’s locked
• box #56 is unlocked by workers #1, #4, #8 and #28, and locked by workers #2, #7, #14 and #56, therefore it’s locked
• box #57 is unlocked by workers #1 and #19, and locked by workers #3 and #57, therefore it’s locked
• box #58 is unlocked by workers #1 and #29, and locked by workers #2 and #58, therefore it’s locked
• box #59 is unlocked by worker #1 and locked by worker #59, therefore it’s locked
• box #60 is unlocked by workers #1, #3, #5, #10, #15 and #30, and locked by workers #2, #4, #6, #12, #20 and #60, therefore it’s locked
• box #61 is unlocked by worker #1 and locked by worker #61, therefore it’s locked
• box #62 is unlocked by workers #1 and #31, and locked by workers #2 and #62, therefore it’s locked
• box #63 is unlocked by workers #1, #7 and #21, and locked by workers #3, #9 and #63, therefore it’s locked
• box #64 is unlocked by workers #1, #4, #16 and #64, and locked by workers #2, #8 and #32, therefore it’s unlocked
• box #65 is unlocked by workers #1 and #13, and locked by workers #5 and #65, therefore it’s locked
• box #66 is unlocked by workers #1, #3, #11 and #33, and locked by workers #2, #6, #22 and #66, therefore it’s locked
• box #67 is unlocked by worker #1 and locked by worker #67, therefore it’s locked
• box #68 is unlocked by workers #1, #4 and #34, and locked by workers #2, #17 and #68, therefore it’s locked
• box #69 is unlocked by workers #1 and #23, and locked by workers #3 and #69, therefore it’s locked
• box #70 is unlocked by workers #1, #5, #10 and #35, and locked by workers #2, #7, #14 and #70, therefore it’s locked
• box #71 is unlocked by worker #1 and locked by worker #71, therefore it’s locked
• box #72 is unlocked by workers #1, #3, #6, #9, #18 and #36, and locked by workers #2, #4, #8, #12, #24 and #72, therefore it’s locked
• box #73 is unlocked by worker #1 and locked by worker #73, therefore it’s locked
• box #74 is unlocked by workers #1 and #37, and locked by workers #2 and #74, therefore it’s locked
• box #75 is unlocked by workers #1, #5 and #25, and locked by workers #3, #15 and #75, therefore it’s locked
• box #76 is unlocked by workers #1, #4 and #38, and locked by workers #2, #19 and #76, therefore it’s locked
• box #77 is unlocked by workers #1 and #11, and locked by workers #7 and #77, therefore it’s locked
• box #78 is unlocked by workers #1, #3, #13 and #39, and locked by workers #2, #6, #26 and #78, therefore it’s locked
• box #79 is unlocked by worker #1 and locked by worker #79, therefore it’s locked
• box #80 is unlocked by workers #1, #4, #8, #16 and #40, and locked by workers #2, #5, #10, #20 and #80, therefore it’s locked
• box #81 is unlocked by workers #1, #9 and #81, and locked by workers #3 and #27, therefore it’s unlocked
• box #82 is unlocked by workers #1 and #41, and locked by workers #2 and #82, therefore it’s locked
• box #83 is unlocked by worker #1 and locked by worker #83, therefore it’s locked
• box #84 is unlocked by workers #1, #3, #6, #12, #21 and #42, and locked by workers #2, #4, #7, #14, #28 and #84, therefore it’s locked
• box #85 is unlocked by workers #1 and #17, and locked by workers #5 and #85, therefore it’s locked
• box #86 is unlocked by workers #1 and #43, and locked by workers #2 and #86, therefore it’s locked
• box #87 is unlocked by workers #1 and #29, and locked by workers #3 and #87, therefore it’s locked
• box #88 is unlocked by workers #1, #4, #11 and #44, and locked by workers #2, #8, #22 and #88, therefore it’s locked
• box #89 is unlocked by worker #1 and locked by worker #89, therefore it’s locked
• box #90 is unlocked by workers #1, #3, #6, #10, #18 and #45, and locked by workers #2, #5, #9, #15, #30 and #90, therefore it’s locked
• box #91 is unlocked by workers #1 and #13, and locked by workers #7 and #91, therefore it’s locked
• box #92 is unlocked by workers #1, #4 and #46, and locked by workers #2, #23 and #92, therefore it’s locked
• box #93 is unlocked by workers #1 and #31, and locked by workers #3 and #93, therefore it’s locked
• box #94 is unlocked by workers #1 and #47, and locked by workers #2 and #94, therefore it’s locked
• box #95 is unlocked by workers #1 and #19, and locked by workers #5 and #95, therefore it’s locked
• box #96 is unlocked by workers #1, #3, #6, #12, #24 and #48, and locked by workers #2, #4, #8, #16, #32 and #96, therefore it’s locked
• box #97 is unlocked by worker #1 and locked by worker #97, therefore it’s locked
• box #98 is unlocked by workers #1, #7 and #49, and locked by workers #2, #14 and #98, therefore it’s locked
• box #99 is unlocked by workers #1, #9 and #33, and locked by workers #3, #11 and #99, therefore it’s locked
• box #100 is unlocked by workers #1, #4, #10, #25 and #100, and locked by workers #2, #5, #20 and #50, therefore it’s unlocked
The red square represents 1, with 2 as the white block immediately to its right and 3 immediately above 2. Then 5 is the white block one space to the left of 3 and 7 the white block one space below 5. Then 11 is the white block right beside 2 and 13 the white block one space above 11. And so on. The primes aren’t regularly spaced on the spiral but patterns are nevertheless appearing. Here’s the Ulam spiral at higher resolutions:
The Ulam spiral x2
The Ulam spiral x4
The primes are neither regular nor random in their distribution on the spiral. They stand tantalizingly betwixt and between. So the numbers represented on this Ulam-like spiral, which looks like an aerial view of a city designed by architects who occasionally get drunk:
Ulam-like spiral of abundant numbers
The distribution of abundant numbers is much more regular than the primes, but is far from wholly predictable. And what are abundant numbers? They’re numbers n such that sum(divisors(n)-n) > n. In other words, when you add the divisors of n less than n, the sum is greater than n. The first abundant number is 12:
Are all abundant numbers even? No, but the first odd abundant number takes a long time to arrive: it’s 45045. The abundance of 45045 was first discovered by the French mathematician Charles de Bovelles or Carolus Bovillus (c. 1475-1566), according to David Wells in his wonderful Penguin Dictionary of Curious and Interesting Numbers (1986):
Here’s the spiral of abundant numbers at higher resolutions:
Abundant numbers x2
Abundant numbers x4
Negating the spiral of the abundant numbers — almost — is the spiral of the deficient numbers, where sum(divisors(n)-n) < n. Like most odd numbers, 15 is deficient:
15 = 3 * 5 → 1 + 3 + 5 = 9 < 15
Here’s the spiral of deficient numbers at various resolutions:
Deficient numbers on Ulam-like spiral
Deficient numbers x2
Deficient numbers x4
The spiral of deficient numbers doesn’t quite negate (reverse the colors of) the spiral of abundant numbers because of the very rare perfect numbers, where sum(divisors(n)-n) = n. That is, their factor-sums are exactly equal to themselves:
Now let’s try numbers n such than sum(divisors(n)) mod 2 = 1 (“n mod 2″ gives the remainder when n is divided by 2, i.e. n mod 2 is either 0 or 1). For example:
There’s not a circle in sight, so you wouldn’t expect to find π amid the pyramids. But it’s there all the same. You can get π from this formula using the square pyramidal numbers:
π from a formula using square pyramidal numbers (Wikipedia)
Here are the approximations getting nearer and near to π:
But there’s something unsatisfying about including 10 as a sum of itself. It’s much more satisfying that 76 can be represented in exactly 76 ways as a sum of distinct primes:
It’s obvious that the sequences are different at each successive step: 1 ≠ 15, 3 ≠ 31, 6 ≠ 48, 10 ≠ 66, 21 ≠ 85, and so on. But seven numbers occur in both sequences: 15, 66, 105, 171, 561, 1326 and 5460. And that’s it — 7 is the 14-th entry in A309507 at the Encyclopedia of Integer Sequences:
I decided to take create graphs of shared numbers in compared sequences like this. In the 135×135 grid below, the brightness of the squares corresponds to the count of shared numbers in the sequence-pair sum(x..x+n) and sum(y..y+n), where x and y are the coordinates of each individual square. I think the grid looks like a city of skyscrapers bisected by a highway:
Count of shared numbers in sequence-pairs sum(x..x+n) and sum(y..y+n)
Note that the bright white diagonal in the grid corresponds to the sequence-pairs where x = y. Because the sequences are identical in each pair, the count of shared numbers is infinite. The grid is symmetrically reflected along the diagonal because, for example, the sequence-pair for x=12, y=43, where sum(12..12+n) is compared with sum(43..43+n), corresponds to the sequence pair for x=43, y=12, where sum(43..43+n) is compared with sum(12..12+n). The scale of brightness runs from 0 (black) to 255 (full white) and increases by 32 for each shared number in the sequence. Obviously, then, the brightness can’t increase indefinitely and some maximally bright squares will represent sequence-pairs that have different counts of shared pairs.
Now try altering the size of the step in brightness. You get grids in which the width of the central strip increases (smaller step) or decreases (bigger step). Here are grids for steps for 1, 2, 4, 8, 16, 32 and 64 (I’ve removed the bright x=y diagonal for the first few grids, because it’s too prominent against duller shades):
12 is an interesting number in a lot of ways. Here’s one way I haven’t seen mentioned before:
12 = 3^1 * 2^2
The digits of 12 represent the powers of the primes in its factorization, if primes are represented from right-to-left, like this: …7, 5, 3, 2. But I couldn’t find any more numbers like that in base 10, so I tried a power flip, from right-left to left-right. If the digits from left-to-right represent the primes in the order 2, 3, 5, 7…, then this number is has prime-power digits too:
I haven’t found any more left-to-right prime-power digital numbers in base 10, but there are more in other bases. Base 5 yields at least three (I’ve ignored numbers with just two digits in a particular base):
When I searched for 81312000 at the Online Encyclopedia of Integer Sequences, I discovered that these are Meertens numbers, defined at A246532 as the “base n Godel encoding of x [namely,] 2^d(1) * 3^d(2) * … * prime(k)^d(k), where d(1)d(2)…d(k) is the base n representation of x.”
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 