The On-Line Blog of Integer Sequences

• 2011-07-11 11 notes

  A066844 - Film speeds

  A066844 - Film speeds

  25, 40, 50, 64, 80, 100, 125, 160, …

  In 1876, Ferdinand Hurter and Vero Charles Driffield founded the field of sensitometry when they established mathematical relationships between the density of silver on film, the sensitivity of that film, and the time needed to develop it. Their measurements were rudimentary, but the relationships (now called H-D curves) are still used by modern sensitometrists.

  The first widely-used film speed standard was the DIN system, first published in 1931. It was based on base 10 logarithms of the sensitivity of the film, multiplied by ten (similar to the decibel scale). A difference in 3° represented approximately a doubling of sensitivity, and an increase of 20° was 100x more sensitive.

  In 1960, the American Standards Association standardized a non-logarithmic film speed scale. In the ASA system, a doubling of the speed was exactly equal to a doubling of sensitivity. Many cameras made during the 1960s and especially the 1970s have both ASA and DIN film speed charts to ease use of film. Finally, in 1987, the DIN and ASA formats were officially merged by the ISO, which supports both an “arithmetic” scale (identical to ASA) and a “logarithmic” scale (identical to DIN). Although use of the logarithmic scale has declined markedly, film must still be officially marked with both — for instance 400/27° or 3200/36°.

• 2011-07-08 9 notes

  A129935 - Numbers n such that ceiling( 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) )

  A129935 - Numbers n such that ceiling( 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) )

  777451915729368, 140894092055857794, 1526223088619171207, …

  Something a little different today. The other day on Reddit’s Math subsite, there was a thread about statements that appeared true for all n less than something large. The first example given was Skewes’ Number, a number where it’s been proven that the number of primes is finally greater than the logarithmic integral. The definition isn’t important, but if you had tested the first hundred quintillion integers (which would take a while) you might have convinced yourself that π(n) < li(n) forever. But it’s not. Since 1933, Skewes’ number has shrunk from the original e^ee79 — it is now somewhere around 10³¹⁶.

  One comment links to a question on Math Overflow with a similar premise. That thread links to several interesting integer sequences in addition to the above. For instance, A102567 are counterexamples to “There is no positive integer n such that the concatenation of n with itself is a square” — the first is 13223140496.

  But the golden egg in those two threads must be a paper called The Strong Law Of Small Numbers by Richard K. Guy. Read through that paper, and try to decide which of them are actually true and which of them have a counter example at some large n.

• 2011-07-06 11 notes

  A038619 - Smallest number that needs more lines when shown in a digital clock than any previous term

  A038619 - Smallest number that needs more lines when shown in a digital clock than any previous term

  1, 2, 6, 8, 10, 18, 20, 28, …

  The seven-segment display was invented far longer ago than probably most people would realize — F.W. Wood patented an eight-segment display in 1908 (there was an extra diagonal segment for a closed 4). Mechanical displays were manufactured throughout most of the twentieth century for gas stations and various other displays, but electrical systems used nixie tubes through the early 1970s. When LED technology became cost-effective, though, seven segment displays became commonplace in numeric contexts everywhere.

  Seven-segment displays aren’t limited only to numbers. Most (although not all) letters in the English alphabet can be unambiguously displayed. Luckily, A-F are all among the representable letters, so such displays can show hexadecimal values without any confusion. Some letters, like M, R, and X, require either some acrobatics, or a fourteen-segment display. As the technology becomes cheaper and cheaper, small dot matrix LCDs are replacing segmented displays almost everywhere.

• 2011-07-01 18 notes

  A007632 - Numbers that are palindromic in bases 2 and 10.

  A007632 - Numbers that are palindromic in bases 2 and 10.

  0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, …

  Palindromic numbers (like their word and sentence counterparts) are numbers that are the same both forwards and backwards. The number of decimal palindromic numbers under any n is well established, and Banks proved in 2004 that almost all palindromes (in any base) are composite. The sum of the reciprocals of all decimal palindromes converges on a constant near 3.37018.

  In any base k > n, the representation of n is a single digit, so all numbers are palindromic in an infinite number of bases. It’s therefore more interesting to look at bases k < n. Charlton Harrison has written programs to search for numbers that are palindromic in both base 10 and base 2. He’s found 120 up through 2¹³⁰. For instance, 585₁₀ = 1001001001₂. As expected, almost all of them are composite; only 6 of them are prime. (The largest prime double-palindrome is 27 digits long!)

• 2011-06-29 15 notes

  A001466 - Denominator of Egyptian fraction for pi - 3

  A001466 - Denominator of Egyptian fraction for pi - 3

  8, 61, 5020, 128541455, …

  While staying in the Thebes region of Egypt in the 1850s, Scottish lawyer and antiquarian Alexander Henry Rhind purchased an ancient six-foot papyrus. First decoded by Hultsch in 1895, the scroll contains dozens of algebraic and geometric techniques and problems. The scroll includes, for instance, an equation for finding cylindrical volume using π ≈ 256/81 ≈ 3.1605. It also includes a table of fractions 2/n for odd n between 5 and 101. The ancient Egyptians’ shorthand for reciprocals led them to prefer to write fractions with non-unit numerators (with the sole exception of 2/3) instead as the sum of unit fractions. These sums have become known as Egyptian fractions.

  The sums are non-trivial. For instance, in the Rhind papyrus’s table, 2/15 is not equivalent to 1/15 + 1/15. Instead, it is 1/10 + 1/30. In 1202, Fibonacci proved that any fraction could be represented in this way as a sum of unit fractions, and even provided an algorithm for calculating them.

  Although four terms are necessary for 2/101, in the 1950s, Paul Erdős and E.G. Straus conjectured that only three terms are necessary for any 4/n. Sierpiński then suggested the same was true for any 5/n. Neither of these conjectures have ever been proven, but no counter-examples have been found for n < 10¹⁴.

• 2011-06-27 51 notes

  A000217 - Triangular numbers

  A000217 - Triangular numbers

  0, 1, 3, 6, 10, 15, 21, …

  Triangular numbers appear in at least two very common games. In bowling, the pins are arranged into rows of 1, 2, 3, and 4 pins each, depicting T₄ = 10. When setting up eight-ball on a billiard table, the triangular rack arranges the balls into T₅ = 15. The largest repdigit triangular number is T₃₆ = 666. Not only is 36 = 6², but it’s also triangular itself (T₈ = 36). All even perfect numbers are triangular. Amazingly, they all have a prime index (i.e. T₇ = 28).

  Fermat’s Last Theorem is only the most well-known case where he claimed to have a solution to a problem but never published it. In 1638, he claimed to have a proof showing that every positive integer was equal to the sum of three triangular numbers, or four square numbers, or five pentagonal numbers, etc. A proof for the triangular case wasn’t published until Gauss did so 130 years after Fermat’s death. Jacobi and Lagrange each independently devised proofs of the square case around the same time. But it wasn’t until 1813 — almost 150 years after Fermat died — that his Polygonal Number Theorem was finally proved in its entirety.

• 2011-06-24 6 notes

  A080601 - Number of positions that the 3 X 3 X 3 Rubik cube puzzle can be in after exactly n moves

  A080601 - Number of positions that the 3 X 3 X 3 Rubik cube puzzle can be in after exactly n moves

  1, 18, 243, 3240, 43239, 574908, …

  The first published algorithms for solving a scrambled Rubik’s Cube were published in 1981 by David Singmaster. The directions included in that book allowed any cube to be solved with fewer than about 80 moves. But the directions weren’t optimized for speed; they were optimized for memorizability. In 1982, Singmaster and Alexander Frey conjectured that the maximum number of moves required to solve any position was in the “low twenties”. Meanwhile, Morwen Thistlethwaite proved that no more than 52 moves were ever necessary.

  For the next few decades, mathematicians and hobbyists pursued the so-called God Number from both ends. The maximum bound was reduced to 42, then 37, then 29. Michael Reid proved in 1995 that a position called Superflip required at least 20 steps to solve. And there the problem sat for ten years. In 2005, Silviu Radu nudged the upper limit down to 28 and then to 27, encouraging more research.

  In 2010, a programmer and three mathematicians teamed up to design a distributed computer simulation. They were able to get Google to contribute the equivalent of 35 CPU-years to run their search. After just a few weeks of solving a covering set of all 43 quintillion valid positions, they announced that they could all be solved with 20 moves or fewer.

  The official world record for solving a Rubik’s cube currently stands at 6.65 seconds.

• 2011-06-22 1 note

  A036057 - Friedman numbers: can be written in a nontrivial way using their digits and the operations + - * / ^ and concatenation of digits (but not of results)

  A036057 - Friedman numbers: can be written in a nontrivial way using their digits and the operations + - * / ^ and concatenation of digits (but not of results)

  25, 121, 125, 126, 127, 128, 153, …

  There are 72 Friedman numbers less than 10000. Of those, only 14 use all of their digits in the proper order. The first few so-called nice Friedman numbers are:

  • 127 = - 1 + 2⁷
  • 343 = (3 + 4)³
  • 736 = 7 + 3⁶
  • 1285 = (1 + 2⁸) * 5 

  Both 123456789 and 987654321 are Friedman numbers, and so is 99999999 (it’s the smallest repdigit Friedman number, equal to (9 + 9/9)^9-9/9 - 9/9). There are so many facts about Friedman numbers, but I won’t put them all here. You should go read about them yourself.

  See also: A020342 - Vampire numbers: n has a nontrivial factorization using n’s digits.

• 2011-06-20 2 notes

  A167213 - The number of ordered ways to achieve a score of n in darts

  A167213 - The number of ordered ways to achieve a score of n in darts

  1, 1, 3, 7, 17, 40, 98, …

  In just about any pub is the United Kingdom, northern Europe, or the United States, there will be a dartboard on the wall. Typically manufactured from braided sisal fibers, the dart board is a disc divided into twenty even sectors and a bulls-eye circle in the center. Each sector has a point value from 1 to 20, although they aren’t placed in order — high values are next to low values, in order to penalize inaccuracies. There’s a “double” ring made up of the outermost part of each sector, and a “triple” ring near the middle of each sector, which score double or triple the point value, respectively.

  The traditional numbering order isn’t ideal — several efforts have been made to come up with improvements. But for various reasons, especially since the most common games played on a dartboard (501 and Cricket) depend less on sheer points and more on hitting specific values, none have ever gained traction.

  Here is a video of John Lowe, three-time world darts champion, scoring the first televised nine-dart finish ever, considered the highest achievement in the game.

• 2011-06-17 10 notes

  A011541 - Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways

  A011541 - Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways

  1729, 4104, 13832, 20683, 32832, 39312, …

  Srinivasa Ramanujan was an Indian mathematician born in Tamil Nadu in 1887. As a child, he showed quick growth in mathematical and analytic abilities. He mastered advanced trigonometry at 12, and invented his own method to solve quartic equations at 15. Two years later, he was calculating Euler’s constant and studying Bernoulli numbers. He struggled through his twenties to keep university scholarships or a job (he was too focused on math).

  In 1913, at the age of 25, he sent a letters to several members of faculty at the University of Cambridge. Only G.H. Hardy recognized his brilliance, and invited him to visit and work with him. For the next several years, it became clear that although Ramanujan’s mind would jump to a solution to a problem, he often struggled to prove that the solution was correct. Nonetheless, his work expanded the fields of mathematical analysis, number theory, infinite series and continued fractions.

  In 1917, Ramanujan fell seriously ill and he recovered in a nursing home for at least a year. He recovered enough by early 1919 to travel back to India, where he fell ill again and died at the tragic age of 32. His singular mind for math is perhaps best represented by a story his friend and mentor Hardy told about a visit while Ramanujan was convalescing:

  Once, in a taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number,” adding that he hoped that wasn’t a bad omen. “No, Hardy,” said Ramanujan, “it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

  There seems to be disagreement about how to generalize the Hardy-Ramanujan number. See also A001235 - Taxi-cab numbers: sums of 2 cubes in more than 1 way.