The On-Line Blog of Integer Sequences

• 2011-06-15 7 notes

  A002313 - Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2

  A002313 - Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2

  2, 5, 13, 17, 29, 37, 41, 53, …

  In 1632, Albert Girard observed that all primes congruent to 1 modulo 4 (meaning that when divided by 4, there is a remainder of 1) could be represented as the sum of two squares. For instance, 17 = 4² + 1². He wasn’t able to prove it, but eight years later, Pierre de Fermat claimed that he had. Fermat never published his proof, and it wasn’t until more than a century later (in 1749) that he Euler provided a full proof in a letter to Christian Goldbach.

  The Eight Queens Puzzle is a challenge proposed in the mid-19th century by chess player Max Bezzel. The puzzle is this: Is it possible the place 8 queens on an 8x8 chess board in such a way that none of them can attack each other. Queens can move any number of spaces in any of the eight orthogonal directions, so no two pieces could share the same row, column, or diagonal. The answer is yes — individual solutions are easy, but Franz Nauck later showed how to build solutions algorithmically, and expanded the puzzle to n queens on an nxn board.

  For n-Queens puzzle where n belongs to the above sequence (n is prime and n = 4m + 1), the two squares can be used to generate a solution. Place a queen at the center. Place the next queen x spaces right and y spaces up, looping from top to bottom and right to left. Repeat until you have placed all of the queens.

• 2011-06-13 25 notes

  A018226 - Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay

  A018226 - Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay

  2, 8, 20, 28, 50, 82, 126

  By the 1940s, the structure of the atom was well-established. At the center, a dense, heavy nucleus of nucleons (protons and neutrons), bound by the powerful strong force. Around the nucleus, there were a number of clouds of electrons (called “shells”) of varying capacities. Electrons are tiny, compared even to the subatomic nucleons. The idea that they would zip around in shells of as many as a dozen or more without interacting with each other was relatively simple to accept.

  But when physicists started investigating the so called nuclear magic numbers, they were presented with a slightly more difficult-to-visualize theory: that there were shells of nucleons within what was believed to be a very densely packed nucleus. The modern nuclear shell model was developed independently by a number of physicists, and it is now almost universally accepted.

  It isn’t, however, the last word in stable nuclei. Observations of the nuclei of the largest elements ever created have revealed non-spherical shells. Beyond 126, deformed nuclei will likely make the next magic number non-linear. The next logical magic number would be 184, but where the next island of stability actually lies is (for now) anyone’s guess.

• 2011-06-10 23 notes

  A101286 - Rounded frequencies in Hz of the notes of the chromatic music scale beginning at A.

  A101286 - Rounded frequencies in Hz of the notes of the chromatic music scale beginning at A.

  55, 58, 62, 65, 69, 73, 78, 82, …

  Although music can technically be composed of notes of any frequency, for a combination of historic and aesthetic reasons, virtually all music sticks to the 12-tone chromatic scale. The scale is logarithmic; two notes with a ratio of frequencies that is a power of 2 is perceived as being similar — the chromatic scale gives them the same name in different octaves. The notes within an octave then are logarithmically spaced. The ratio of a tone’s frequency and the next higher tone’s frequency is 2^1/12 ≈ 0.0833. This is known as a semitone or half-step.

  But what is the frequency of a specific note, say, the A above middle C? You could tune your guitar (or piano or any other instrument) to any frequency, and as long as the tuning is self-consistent, the music would sound fine to anyone without perfect pitch. Throughout the Middle Ages, little attention was paid to tuning, so you could find pipe organs in the same city that were off by five or more semitones. By the 18th century, tuning forks were being used to tune some instruments. But since frequency-measuring equipment was inexact, guaranteeing two tuning forks were the exact same tone was essentially impossible. In addition, there were historical periods of “pitch inflation”. A higher-tuned string instrument sounds louder than others (due to the tension on the strings), so orchestras would over time tune higher and higher in order to sound more “brilliant” and louder.

  The French government legally set the A above middle C as 435 Hz in 1859. Others were attracted to the mathematical purity of middle C itself equal to 2⁸ = 256 Hz, which made A = 430.54 Hz. The British attempted to standardize, but couldn’t settle on their own value. In 1939, an international standard was finally adopted: A = 440 Hz. But several orchestras (in particular the New York Philharmonic and Boston Symphony Orchestra) insist on ignoring the standard concert pitch. They use 442 Hz as their tuning point.

• 2011-06-08 7 notes

  A099009 - List of fixed points of the Kaprekar mapping

  A099009 - List of fixed points of the Kaprekar mapping f(n) = n’ - n”, where in n’ the digits of n are arranged in descending, in n” in ascending order.

  0, 495, 6174, 549945, 631764, 63317664, …

  The Kaprekar mapping (or function, or routine — there doesn’t seem to be a single name) is a simple mathematical trick discovered in 1949 for 4-digit numbers. Take a 4-digit number (let’s say, n = 7912). Arrange the digits in descending order (n’ = 9721) and ascending order (n”=1279). Now subtract (K(n) = n’ - n” = 9721 - 1279 = 8442). Repeat until you hit a cycle. In this case, 8442 -> 5994 -> 5355 -> 1998 -> 8082 -> 8532 -> 6174 -> 6174. Start over with another number. And another. Every time, you end up at either 6174 or 0. The number 6174 is sometimes called the “kernel” of this process.

  The routine has been generalized for k-digit numbers. All 1- and 2-digit numbers lead to zero. All 3-digit numbers end up at 495 or 0. All 5-digit numbers lead not to a single number but to a one of three possible cycles. Malcolm Lines found two kernels each for 6-, 8-, and 9-digit Kaprekar mappings, and no kernel at all for 7-digit.

  If you take any Kaprekar process kernel and add up all of the digits (6+1+7+4, for instance) and repeat until you have a single digit, you will always end up with 9.

• 2011-06-06 10 notes

  Answer to Integer Sequence Puzzle #8

  This is the answer to the Integer Sequence Puzzle posted on Saturday.

  A028355 - How the astronomical clock (“Orloj”) in Prague would strike 1,2,3,…,24,25,..

  1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, …

  On the Old Town Square in Prague, there stands a 600 year old astronomical clock. Built and updated more than once during the 15th century, the Prague Orloj doesn’t just tell the time. It shows sunrise and sunset times, Zodiac signs, the Sun and Moon’s locations on the ecliptic, and the time in the old Czech time scale (number of hours since sunset).

  The hourly chimes are designed in an interesting way. The mechanism strikes, 1, 2, 3, or 4 times. And it always moves in a pattern from 1 to 4, then back down towards 1. Grouping these sets of chimes allows it to strike any hour without having to backtrack or skip (3 then 2 is 5; after that comes 1, 2, and 3, which is 6).

  Even by today’s standards, the clock is a work of art and a technical masterpiece. After suffering damage in World War II, the clock was rebuilt and today functions as well as ever. For its 600th birthday in 2010, Prague put on a spectacular light show.

• 2011-06-04 2 notes

  Integer Sequence Puzzle #8

  Try to identify this integer sequence without searching OEIS.

  1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, …

  I will post the answer on Monday.

• 2011-06-03 9 notes

  A076336 - (Provable) Sierpinski numbers

  A076336 - (Provable) Sierpinski numbers

  78557, 271129, 271577, 322523, 327739, …

  In 1960, Wacław Sierpiński proved that there were infinitely many odd natural numbers k for which k2ⁿ + 1 was composite for every positive value of n. Two years later, John Selfridge proved that 78557 met this criteria (he was able to show that for any value of n, 78557 x 2ⁿ + 1 had at least one of a set of five prime factors). For the next five years, Selfridge and Sierpinski worked to find other numbers of this form, but could not prove than any lesser numbers met the criteria. In 1967, they conjectured that 78557 was the smallest.

  By March 2002, there were 17 numbers smaller for which their status was unknown. Louis Helm at the University of Michigan and David Norris at the University of Illinois decided that instead of wait for proofs, they would start a distributed search for counter-examples. If they could find an n for all 17 candidates for which k2ⁿ + 1 was prime, they would prove the conjecture.

  Over the past nine years, the project has narrowed the list to 6 candidates. You can join the search and help close the book on an unsolved problem in number theory.

• 2011-06-01 39 notes

  A001676 - Number of h-cobordism classes of smooth homotopy n-spheres

  A001676 - Number of h-cobordism classes of smooth homotopy n-spheres

  1, 1, 1, 1?, 1, 1, 28, 2, …

  Imagine you’re a microscopic person on the surface of a 3-dimensional sphere. If you walk far enough in any direction, you’ll eventually end up back where you started. Now draw a loop anywhere on the surface of the sphere. Shrink that loop and keep shrinking it until it’s a point. No matter where you drew that loop on the sphere, it would shrink down to a point. Imagine doing the same things to a torus. If you walked in any direction, you’d end up back where you started, so you can’t identify it as a non-sphere that way. But there are ways to draw loops on the surface that can never be shrunk to a point. A sphere (and shapes topologically homeomorphic to it) is the only simply connected closed 2-manifold (2 because the important bit is the surface, and it’s 2-dimensional).

  Now imagine a 4-dimensional sphere with a 3-dimensional surface. (Yes, this is harder.) You can shrink any loop to a point. That has been known for a long time. But are there any shapes not homeomorphic to a 3-sphere for which this is also true? In 1900, Henri Poincaré (pwan-kar-AY) created this “shrinking a loop to a point” technique (called fundamental groups) and asked if it was sufficient to identify a 3-sphere. More than a hundred years later, reclusive Russian mathematician proved that Poincaré’s conjecture was correct, refused the Fields Medal (analogous to the Nobel Prize for mathematics), and refused a million dollar prize.

  The Generalized Poincaré Conjecture has been proved (or disproved) in all dimensions except for 4. Is a closed 4-dimensional manifold homeomorphic to a 4-sphere? No one’s sure, yet. In fact, 4-dimensional space is a pretty chaotic place.

• 2011-05-30 1 note

  Answer to Integer Sequence Puzzle #7

  This is the answer to the Integer Sequence Puzzle posted on Saturday.

  A006052 - Number of different magic squares of order n that can be formed from the numbers 1, …, n^2

  1, 0, 1, 880, 275305224

  The magic square was known to Chinese mathematicians sometime before the year 650 BC. The Lo Shu magic square is the unique 3×3 square (ignoring reflections and rotations) that can be filled with the numbers from 1 to 9 such that every row, column, and diagonal adds up to the same number (in the case of n=3, 15).

  

  Squares for n=4 were known at least a thousand years ago in India, and squares for n=5 and n=6 were discovered around the same time in Baghdad. Squares where n is odd or a multiple of 4 are relatively easy to discover. Those where n is even but not a multiple of 4 are significantly harder, and methods other than trail and error were not identified until the 20th century. The 880 squares of order 4 were enumerated by de Bessy in 1693.

  The number of possible 6×6 magic squares is unknown, though Pinn and Wieczerkowski narrowed it to (1.7745 ± 0.0016) × 10¹⁹ with computer simulations in 1998.

• 2011-05-28 0 notes

  Integer Sequence Puzzle #7

  Try to identify this integer sequence without searching OEIS.

  1, 0, 1, 880, 275305224

  I will post the answer on Monday. Next week will be yet another theme week, focusing on series that have unknown elements. For this puzzle, for instance, there is a next number in the sequence, but its value is unknown (although a 0.1%-wide window has been narrowed).