The On-Line Blog of Integer Sequences

A000602 - Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers

Wed, 05 Oct 2011 11:38:54 -0400

A000602 - Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers

1, 1, 1, 1, 2, 3, 5, 9, …

The chemical we call alcohol is actually more specifically ethyl alcohol (or ethanol). There are other alcohols: methyl alcohol (wood alcohol, similarly intoxicating but poisonous) and isopropyl alcohol (rubbing alcohol) are two of the most common. The simplest alcohols are strings of carbon atoms with a hydroxyl group attached to one of the carbons. In the case of methyl, n = 1, a single carbon atom, has hydroxyl attached, and the three other spots (of the four valences that allow carbon to attach to other atoms) are taken up by hydrogen. Ethyl, n = 2, has two carbons.

Once you get beyond n = 3 (propyl alcohol), the branching behaviors of the four valences in carbon atoms become apparent. Alkane (similar to alcohols, but without the hydroxyl group) examples abound. Butane and isobutane are the two isomers for n = 4. Hexane (n = 6) comes in five different common shapes with sometimes very different properties. Decane (n = 10) has 75 isomers, all of which are flammable liquids.

A084617 - Maximum number of circles with diameter 1 that can be packed in a square of side length n

Mon, 03 Oct 2011 09:21:18 -0400

A084617 - Maximum number of circles with diameter 1 that can be packed in a square of side length n

1, 4, 9, 16, 25, 36, 49, 68, …

Area packing is an interesting field of study. The problems are easy to define to a layperson: How many circles can fit into a square of a given area? What’s the smallest square some number of smaller squares can fit into? Everyone has packed a box or the trunk of a car and can understand the challenges and techniques for packing a set of objects into a given space.

But the simplicity of understanding the field belies the complexity of the mathematics. The possibilities grow much faster than the number of objects you’re packing. For packing circles in a square, values above about a 5x5 square are conjectures. For packing squares inside the smallest square possible, the best packing for as few as 11 objects has still not been proven.

A139250 - Toothpick sequence

Fri, 30 Sep 2011 00:36:05 -0400

A139250 - Toothpick sequence

0, 1, 3, 7, 11, 15, 23, 35, 43, …

Created by Omar Pol, the toothpick sequence is defined as follows:

At t = 0, there are no toothpicks
At t = 1, there is one toothpick, placed vertically on a plane
To grow from t = n to t = n + 1, place new toothpicks with each of their midpoints touching one endpoint that does not touch another toothpick. All toothpicks must be perpendicular to the toothpick it is touching.

Here is an animation created by David Applegate of t <= 32:

He also created a JavaScript-powered tool that allows you to play with various conditions and watch the toothpick sequence grow.

A116369 - Day of the week corresponding to Jan 01 of a given year

Wed, 28 Sep 2011 00:36:05 -0400

A116369 - Day of the week corresponding to Jan 01 of a given year

7, 2, 3, 4, 5, 7, 1, 2, 3, 5, 6, 7, …

The only difference between the Gregorian calendar (introduced in 1582 by Pope Gregory XIII) and its predecessor the Julian calendar (introduced in 45 BC by Julius Caesar) is their leap year rules. The older calendar had a leap year every four years, period. The newer one skipped leap years every hundred (except every 400th year did have one). Oh, one other minor difference: ten days in 1582 never happened; in Catholic countries that switched immediately, October 4th, 1582 was followed by October 15th, 1582. In other countries, different dates were skipped (the British Empire didn’t change to the Gregorian calendar until 1752, so for 170 years it was a different date in Britain than it was in France).

But with the sole exception of years in which switchovers happened, for the past two thousand years, there have only been fourteen possible annual calendars. Every year starts on one of the seven weekdays, and every year is either a leap year or it is not. And there is a pattern to the order of the calendars. Although a true permanent calendar would need a 400-year cycle, in practice you can get away with just 28 years (the lowest common multiple of the 7-day week and the 4-year leap year) until 2100 screws it up.

A003418 - a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, ..., n}

Fri, 26 Aug 2011 08:51:00 -0400

A003418 - a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, …, n}

1, 1, 2, 6, 12, 60, 60, …

The following is a guest post contributed by Brian Ingalls:

I can’t remember how it came up, but I was talking with some co-workers at a bar the other day about tontines. You may not know them by name, but you might know the episode of The Simpsons called Raging Abe Simpson and His Grumbling Grandson in “The Curse of the Flying Hellfish”. In this episode, Grandpa Simpson was part of a unit in WWII that “found” a bunch of art while raiding a Nazi mansion. They take the art, seal it up, bury it, and form an agreement, that the last member of the unit left alive will own the artwork.

While having this discussion, I looked up tontines on Wikipedia. I learned that traditionally, a tontine is “an investment scheme for raising capital”. This makes it much more desirable to participate, because you actually get a return on your investment before you are a hundred years old and the last remaining member. Each member of the tontine owns an even share in the investment, and they all get dividends. When someone dies, their share is evenly divided among the remaining members. In this version however, the initial capitol is never returned, and when the final member dies, the agreement is dissolved.

This got me thinking about what the most effective means for dividing up the initial shares so that the share division process is as easy as possible. I realized that the LCM would be the answer. A number of shares so that when each member dies, their shares are evenly divisible amongst the remaining members.

A077586 - 2^(2^p-1)-1

Fri, 12 Aug 2011 14:23:53 -0400

A077586 - 2^(2^p-1)-1

7, 127, 2147483647, …

A double Mersenne number is a Mersenne number n = 2^k-1 where k is a Mersenne prime. Since the first few Mersenne primes are k = 3, 7, and 31, the first double Mersenne numbers are those shown above. The next one is 2¹²⁷-1, and after that is 2²⁰⁴⁷-1. They quickly get very large.

But not all Mersenne numbers with a prime exponent are necessarily prime — that’s a necessary condition, but not a sufficient one. The first four double Mersenne numbers are, but the next three (M_M₁₃, M_M₁₇, M_M₁₉, M_M₃₁) are not. The next double Mersenne number, M_M₆₁, is certainly a candidate, but it’s got 7 × 10¹⁷ digits, and no existing factoring method has yet been able to find any factors.

A distributed search for factors of higher double Mersenne number has been started, but due to the size of the numbers, it may be a very long time indeed until any of them are proven to be prime.

A006877 - In the `3x+1' problem, these values for the starting value set new records for number of steps to reach 1

Mon, 01 Aug 2011 00:00:00 -0400

A006877 - In the `3x+1’ problem, these values for the starting value set new records for number of steps to reach 1

1, 2, 3, 6, 7, 9, 18, 25, …

In 1937, Lothar Collatz described an operation on positive integers. If the integer is even, divide it by 2. If it is odd, triple it and add one. And then he posed a problem, which has since been called the Collatz conjecture (and a dozen other less common names) — if you apply this operation repeatedly, will it always eventually reach 1, no matter what number you start with?

The most promising avenues of proof are by disproving the alternatives. If the Collatz conjecture were false, that would mean that one of the following alternatives were true:

For some odd n, there is an infinitely divergent series, i.e. ~~3n+1 is odd, and 3(3n+1)+1 is odd, and so forth forever,~~ it never equals 2^k for any k, or
There exists a cycle which does not include 1

Tim Conway and Jeffrey Lagarias each examined the latter alternative through the 70s and 80s, Lagarias proving that there are no cycles of length < 275,000. More recently, several researchers (most recently Oliveira e Silva) have searched for some n that does not have a finite path to 1. So far, no counter-examples are known through at least 5 × 10¹⁸.

Unfortunately, the Collatz Conjecture may never be proved. Paul Erdős himself once said “mathematics is not yet ready for such problems” in reference to this problem. He may have been right — In 1972, Conway proved that general Collatz-type problems can be formally undecidable.

A033623 - Grundy function for turn-at-most-4-coins game.

Wed, 27 Jul 2011 08:25:32 -0400

A033623 - Grundy function for turn-at-most-4-coins game.

1, 2, 4, 8, 15, 16, 32, 51, …

Impartial games are those where there is no hidden information, no element of chance, and you don’t need to know whose turn it is to determine the optimal move. Tic-tac-toe and chess are not impartial games. In those cases, since each player has her own pieces (X’s, or white), the optimal move for one player is not always the same as that for the other. The simplest (and in some ways, canonical) impartial game is Nim. The game starts with several piles of tokens. Players alternate turns taking one or more tokens from any one pile. Typically, the player who takes the last token loses (this type of impartial game is called “misère”), but the rule can be flipped (“normal play”).

The Sprague-Grundy theorem (discovered independently by two mathematicians in the 1930s) shows that normal play impartial games of any sort are always equivalent to a game of Nim of sufficient size. By mapping the game states of a game to a directional graph and then assigning values (called nim values or nimbers) to the game states, you can determine the optimal play for any normal play impartial game.

A140267 - Nonnegative integers in balanced ternary representation (with 2 standing for -1 digit)

Fri, 22 Jul 2011 14:39:00 -0400

A140267 - Nonnegative integers in balanced ternary representation (with 2 standing for -1 digit)

0, 1, 12, 10, 11, 122, 120, …

In number bases that most people are familiar with (decimal, binary, octal, etc), all of the digits represent positive values. There is no digit that could appear in the ones place that would represent a negative value. You would have to place a minus symbol before the number. In signed-digit representations, however, there are digits that represent negative numbers. Balanced forms are a special subset of signed-digit representations where for a base k, there are digits for all values between -⌊k/2⌋ and ⌊k/2⌋. Balanced ternary, for instance, has values for -1, 0, and 1. (These are usually written as +, 0, and -).

At first, this can be confusing. Like standard ternary, it is easy to represent powers of 3:

1 = 3⁰ = +
3 = 3¹ = +0
9 = 3² = +00

Numbers one less than a power of three have a - digit in the ones position:

2 = 3¹ - 1 = +-
8 = 3² - 1 = +0-

Other numbers can be constructed with a little bit more thought.

But of what use is balanced ternary? The most valuable use is in fast integer arithmetic. The single-digit addition tables for balanced ternary requires 33% fewer carries. And multiplication tables need no carried digits at all. Some fast math libraries are therefore implemented with balanced ternary, and CPUs that use balanced ternary at the circuitry level can be simpler, faster, and require less power. (Hypothetically, two-pan balances and monetary systems could be improved with a balanced ternary system, but the complexities of doing power-of-three math in ones head probably means this will never be implemented.)

A105386 - Morse code alphabet where "." = 0 and "-" = 1.

Wed, 13 Jul 2011 13:52:48 -0400

A105386 - Morse code alphabet where “.” = 0 and “-” = 1.

1, 8, 10, 4, 0, 2, 6, 0, …

The electrical telegraph was not the first method for transmitting information over long distances. For centuries, a number of non-electrical visual communication mechanisms were in wide use. Claude Chappe’s semaphore network was widely used in France for most of the first half of the 19th century, especially by Napoleon to coordinate his armies. The Prussian semaphore system transmitted instructions hundreds of miles for several decades. And smoke signals and reflected light signals have been used by indigenous peoples since ancient times.

When the first electrical telegraph was invented by Samuel Thomas von Sömmering in 1809, it used up to 35 different wires — one for almost every letter and number. Gauss and Weber’s telegraph was the first to be used for regular communication, and although it encoded letters onto two wires, they chose to use a system of binary representation: a current in one direction was a 0, in the opposite direction was a 1. Samuel Morse’s electric telegraph was the first commercially viable system in the United States, and the encoding system that he and/or Alfred Vail invented made it fast and simple to send text messages.

The telegraph spread wildly. In 1838, he sent the first telegram a distance of 2 miles. In 1844, he sent a message from Washington, DC to Baltimore (about 40 miles). By 1861, the telegraph stretched from coast to coast and the Pony Express was abandoned as a result. In 1966, the first trans-Atlantic telegraph line was laid, and in 1902 cables also stretched across the Pacific, encircling the world in telegraphy.

A066844 - Film speeds

Mon, 11 Jul 2011 12:55:46 -0400

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°.

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

Fri, 08 Jul 2011 09:18:31 -0400

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^{e^e⁷⁹} — 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.

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

Wed, 06 Jul 2011 12:20:57 -0400

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.

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

Fri, 01 Jul 2011 10:31:00 -0400

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!)

A001466 - Denominator of Egyptian fraction for pi - 3

Wed, 29 Jun 2011 00:00:06 -0400

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.

A000217 - Triangular numbers

Mon, 27 Jun 2011 00:00:06 -0400

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.

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

Fri, 24 Jun 2011 12:14:47 -0400

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.

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

Wed, 22 Jun 2011 14:59:33 -0400

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.

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

Mon, 20 Jun 2011 09:00:00 -0400

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.

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

Fri, 17 Jun 2011 08:52:18 -0400

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.