The On-Line Blog of Integer Sequences

A258107 - Smallest number > 1 whose representation in all bases up to n consists only of zeros and ones

2, 3, 4, 82000

The representation of n=5, 82000:

In base 2: 10100000001010000

In base 3: 11011111001

In base 4: 110001100

In base 5: 10111000

A000215 - Fermat numbers: 2^(2^n) + 1, n >= 0

3, 5, 17, 257, 65537, 4294967297, ...

In 1650, Pierre de Fermat conjectured that numbers of the form Fn = 22n+1 were prime. But these numbers get large quickly (n=5 has ten digits), and it wasn't until 1732 that Euler factored F5 = 641 * 6700417. In 1855, n = 6 solved, and through the middle of the 20th century, several factors of other Fermat numbers were found.

When computers took over, this work happened faster, but still not quickly. As of today, only seven Fermat numbers are completely factored. The largest of those, F11 has only five prime factors, and one of those is 560 digits long, and none have been finished since 1990.

But there still several open questions. Are there any prime Fermat numbers for n > 4? If there are, are there an infinite number of them? Richard Guy and John Conway made a bet in 1996 that no Fermat numbers would be completely factored in the next 20 years -- and Conway seems likely to collect.

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.

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.

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:

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).

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.

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

7, 127, 2147483647, ...

A double Mersenne number is a Mersenne number n = 2k-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 (MM13, MM17, MM19, MM31) are not. The next double Mersenne number, MM61, is certainly a candidate, but it's got 7 × 1017 digits, and no existing factoring method has yet been able to find any factors.

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 2k 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.

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").

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.

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.

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.

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 eee79 -- 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.

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.

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.

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.