The Connoisseur of Number SequencesFor more than 50 years, the mathematician Neil Sloane has curated the authoritative collection of interesting and important integer sequences.

By: Erica KlarreichAugust 6, 2015

Neil Sloane is considered by some to be one of the most influential mathematicians of our time.

That’s not because of any particular theorem the 75-year-old Welsh native has proved, though over the course of a more than 40-year research career at Bell Labs (later AT&T Labs) he won numerous awards for papers in the fields of combinatorics, coding theory, optics and statistics. Rather, it’s because of the creation for which he’s most famous: the Online Encyclopedia of Integer Sequences (OEIS), often simply called “Sloane” by its users.This giant repository, which celebrated its 50th anniversary last year, contains more than a quarter of a million different sequences of numbers that arise in different mathematical contexts, such as the prime numbers (2, 3, 5, 7, 11 … ) or the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13 … ). What’s the greatest number of cake slices that can be made with n cuts? Look up sequence A000125 in the OEIS. How many chess positions can be created in n moves? That’s sequence A048987. The number of ways to arrangen circles in a plane, with only two crossing at any given point, is A250001. That sequence just joined the collection a few months ago. So far, only its first four terms are known; if you can figure out the fifth, Sloane will want to hear from you.A mathematician whose research generates a sequence of numbers can turn to the OEIS to discover other contexts in which the sequence arises and any papers that discuss it. The repository has spawned countless mathematical discoveries and has been cited more than 4,000 times.

“Many mathematical articles explicitly mention how they were inspired by OEIS, but for each one that does, there are at least ten who do not mention it, not necessarily out of malice, but because they take it for granted,” wrote Doron Zeilberger, a mathematician at Rutgers University.

Courtesy of Neil Sloane

The number of ways to arrange ncircles in a plane, with only two crossing at any given point, is sequence A250001 in the OEIS.

The collection, which began in 1964 as a stack of handwritten index cards, gave rise to a 1973 book containing 2,372 sequences, and then a 1995 book, co-authored with mathematicianSimon Plouffe, containing just over 5,000 sequences. By the following year, so many people had submitted sequences to Sloane that the collection nearly doubled in size, so he moved it onto the Internet. Since then, Sloane has personally

created entries for more than 170,000 sequences. Recently, however, he’s had help processing the torrent of submissions he receives each year from all over the world: Since 2009 the collection has been run as a wiki, and it now boasts more than 100 volunteer editors.But the OEIS is still very much Sloane’s baby. He spends hours each day vetting new submissions and adding sequences from archived papers and correspondence.

Quanta caught up with Sloane over Skype last month as he sorted through sequences in his attic home office in Highland Park, N.J. Formerly a children’s playroom, its garish wallpaper is tempered by giant stacks of papers, and, as Sloane put it, “enough computers so I don’t need a heater.” An edited and condensed version of the interview follows.QUANTA MAGAZINE: Tell me how you started the OEIS. Some sequences came up in your research as a graduate student, right?NEIL SLOANE: It was my thesis. I was looking into what are now called neural networks. These are networks of [artificial] neurons, and each neuron fires or doesn’t fire and is connected to other neurons which fire or don’t fire depending on the signal. I wanted to know whether the activity in some of these networks was likely to die out or keep firing.Some of the simplest cases gave rise to sequences. I took the simplest one and, with some difficulty, worked out half a dozen terms. [It] goes 1, 8, 78, 944…. I needed to know how fast it grew, and I looked it up in the obvious places, and it wasn’t there.

I started making a collection of sequences, so the next time this came up, I’d have my own table to look up. I made a little collection of file cards, and then they became punched cards and then magnetic tape and eventually the book in 1973.

And when did you start sharing your collection with other people?Oh, right away. I mean, within a year or two. The word got around, and you know, letters started coming in. And as soon as the book came out, there was a flood of letters. I’m still going through the binders from that period. The project [now] is to sort through all the interesting documents from the past, which now goes back 51 years. A lot of them are in binders. A lot of them are not, unfortunately. Over there, there’s about an eight- or nine-feet stack of papers that haven’t been sorted.

It’s very slow work. I have to go through these 50 binders and figure out what’s worth scanning, what’s worth preserving, what is available online so we don’t need to scan it. But I’m also finding lots of new sequences as I go along, that for one reason or another I didn’t include the first time around.

Besides the books about sequences, you’ve also co-authored two guidebooks to rock climbing in New Jersey.I did it with my climbing partner, Paul Nick. We spent a lot of time driving around New Jersey climbing on crags and taking photographs and collecting route information. There

were a lot of restrictions. A lot of cliffs were on private property, so we couldn’t officially include them in the book.

Do you have any favorite mathematical discoveries that came about because of the OEIS?One of the most famous discoveries has to do with a formula discovered by Gregory, an astronomer back in Newton’s day, for π/4. The formula says that π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 and so on. It’s a good way of computing π if you don’t have any better way. So somebody did this, but wondered what would happen if you stopped after a while. So he truncated the sum after 500,000 terms and looked at the number, and he worked it out to many decimal places. He noticed, of course, that it was different from π.

John Smock for Quanta Magazine

Sloane is discovering new integer sequences in unsorted stacks of documents collected over 51 years.

He looked at where it differed, and it differed after five decimal places. But then it agreed for the next ten places, and then it disagreed for two decimal places. Then it agreed for the next ten places, and then it disagreed. This was absolutely amazing, that it would agree everywhere except at certain places.

Then I think it was Jonathan Borwein who looked at the differences [between π and the truncated sum]. When you subtract you get a sequence of numbers, and he looked it up in the OEIS, and it wasn’t there. But then he divided by 2 and looked it up, and there they were. It was sequence A000364. It was the Euler numbers.He and his two collaborators studied this, and they ended up with a formula for the error term. If you truncate Gregory’s series after not just 500,000 terms, but after n terms, wheren can be anything you want, you can give an exact formula for the error.It was absolutely miraculous that this was discovered. So, it’s a theorem that came into existence because of the OEIS.

Tell me about some sequences you like. What makes a sequence appealing to you?It’s a bit like saying, “What makes a painting appealing?” or “What makes a piece of music appealing?” In the end, it’s just a matter of judgment, based on experience. If there is some rule for generating the sequence which is a bit surprising, and the sequence turns out to be not so easy to understand, that makes it interesting.

There’s a sequence of Leroy Quet’s which produces primes. It chugs along, but it’s like Schrödinger’s cat; we don’t know if it exists [as an infinitely long sequence] or not. I think we’ve computed 600 million terms, and so far it hasn’t died. It would be nicer — or maybe it would be less nice — if we could actually analyze it.

How often do you get a new sequence that makes you say, “I can’t believe no one has ever thought of this before”?This happens all the time. There are many gaps, even now. I fill in these gaps myself quite often when I come across something in one of these old letters. We’re a finite community. It’s easy to overlook even an obvious sequence.

To what extent is there a clear aesthetic about which sequences deserve to be in the OEIS?We have arguments about this, of course, because somebody will send in a sequence that he or she thinks is wonderful, and we the editors, look at it and say, “Well, that’s really not very interesting. That’s boring.” Then the person who submitted it may get really annoyed and say, “No, no, you’re wrong. I spent a lot of time on this sequence.” It’s a matter of judgment, and in the end I have the final say. Of course, I’m very influenced by the other editors-in-chief.

One of our phrases is, “This is too specialized. This is too arbitrary. This is not of general interest.” For instance, primes beginning with 1998 would not be so interesting. Too specialized, too arbitrary, so that would be rejected.

It might not be rejected if it had been published somewhere — if it was on a test, say. We like to include sequences that appear on IQ tests. It’s always been one of my goals to help people do these silly tests.

One of the features on the OEIS is the option to listen to a sequence musically. What do you think that adds?Well, it’s another dimension of looking at the sequence. Some sequences, you get a good feeling for them by listening to them. Some of the sequences almost sound like music. Others just sound like rubbish.

You’ve said that you think Bach would have loved the OEIS.I think music is very mathematical, obviously, and so he would have appreciated the OEIS. He would have understood it. He probably would have joined in, contributed some sequences. Maybe he would have composed some pieces that we could use.

Do you have a sense of the magnitude of the OEIS’ impact?Not really. I know it’s helped a lot of people, and it’s very famous. We have sequence fans from all over the world. You’ll see many references from unexpected places to the OEIS: journals, books, theses from civil engineering or social studies that mention sequences. They come up all over the place.

Are there other repositories of mathematical information that you wish existed, but don’t yet?You would like an index to theorems, but it’s hard to imagine how that would work.

We’re trying to get a collaboration going with the Zentralblatt — the German equivalent of Math Reviews’ MathSciNet — about making it possible to search for formulas in the OEIS. Suppose you want the summation of xn over n2 + 3, where the sum goes from one to infinity. It’s very hard to look that up in the OEIS at present.You’re retired from AT&T Labs, but looking at your list of recent publications and your activity with the OEIS, you seem anything but retired.I have an office at Rutgers, and I give lectures there, and I have students, and I’m even busier back here in my study running the OEIS and doing research and going around the world giving talks and so on. I’m busier than ever.

There are more than 4,000 people registered on the OEIS website. They range from professional mathematicians to recreational mathematicians, right?A child just registered the other day, and said, “I’m ten years old, and I’m very smart.” So it’s a wide-ranging group of people all over the world, from many different occupations. One of the things people like about the OEIS is this opportunity to collaborate, to exchange emails with professionals. It’s one of the few opportunities that most people have to talk to a real mathematician.

Do you feel that there is a divide between “serious mathematics” and “recreational mathematics”? Or do you tend not to think in those terms?I don’t think in those terms. I don’t think there’s much difference. If you look hard enough, you can find interesting mathematics anywhere.

A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1. (Formerly M1100 N0419)

62

1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235,

15226 (list; graph; refs; listen; history; text; internal format)OFFSET 0,2

COMMENTS Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical interpretation: Define a number of planes in space to be in general arrangement when

(1) no two planes are parallel,(2) there are no two parallel intersection lines,(3) there is no point common to four or more planes.Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement. (See the comments on A000124 for general arrangement with

lines.) These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is A000124(n-1). Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n)=a(n-1)+A000124(n-1). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X which have no exactly one element in common with Y. - Milan Janjic, Dec 28 2007

a(n) is the number of compositions (ordered partitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 23 2009

{a(k): 0 <= k < 4} = divisors of 8. - Reinhard Zumkeller, Jun 17 2009a(n) is also the maximum number of different values obtained by summing n consecutive positive integers with all possible 2^n sign combinations. This maximum is first reached when summing the interval [n, 2n-1]. - Olivier Gérard, Mar 22 2010

a(n) contains only 5 perfect squares > 1: 4, 64, 576, 676000, and 75203584. The incidences of > 0 are given by A047694. - Frank M Jackson, Mar 15 2013

Given n tiles with two values - an A value and a B value - a player may pick either the A value or the B value. The particular tiles are [n, 0], [n-1, 1], ..., [2, n-2] and [1, n-1]. The sequence is the number of different final A:B counts. For example, with n=4, we can have final total [5, 3] = [4, _] + [_, 1] + [_, 2] + [1, _] = [_, 0] + [3, _] + [2, _] + [_, 3], so a(4) = 2^4 - 1 = 15. The largest and smallest final A+B counts are given by A077043 and A002620 respectively. -Jon Perry, Oct 24 2014

REFERENCES V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_3.

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.

A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964)

LINKS T. D. Noe, Table of n, a(n) for n=0..1000A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

Svante Linusson, The number of M-sequences and f-vectors, Combinatorica, vol 19 no 2 (1999) 255-266.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du

Québec à Montréal, 1992.D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.

L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, 2014.Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

H. P. Robinson, Letter to N. J. A. Sloane, Aug 16 1971, with attachmentsEric Weisstein's World of Mathematics, Cake NumberEric Weisstein's World of Mathematics, Cube Division by PlanesEric Weisstein's World of Mathematics, Cylinder CuttingEric Weisstein's World of Mathematics, Space Division by PlanesR. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller, Jun 17 2009]

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA a(n) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.G.f.: (1-2*x+2x^2)/(1-x)^4; - [Simon Plouffe in his 1992 dissertation.]E.g.f.: (1+x+x^2/2+x^3/6)*exp(x).a(n) = binomial(n,3)+binomial(n,2)+binomial(n,1)+binomial(n,0). [Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006]

Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson, Oct 23 2007

EXAMPLE a(4)=15 because there are 15 compositions of 5 into four or fewer parts. a(6)=42 because the sum of the first four terms in the 6th row of Pascal's triangle is 1+6+15+20=42. - Geoffrey Critzer, Jan 23 2009

For n=5, (1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 35) and their opposite are the 26 different sums obtained by summing 5,6,7,8,9 with any sign combination. -Olivier Gérard, Mar 22 2010

MAPLE A000125 := n->(n+1)*(n^2-n+6)/6;

MATHEMATICA Table[(n^3+5n+6)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* Harvey P. Dale, Jan 19 2013 *)

PROG (PARI) a(n)=(n^2+5)*n/6+1 \\ Charles R Greathouse IV, Jun 15 2011(MAGMA) [(n^3+5*n+6)/6: n in [0..50]]; // Vincenzo Librandi, Nov 08 2014

CROSSREFS Cf. A000124, A003600.Bisections give A100503, A100504.Row sums of A077028.A005408, A000124, A016813, A086514, A058331, A002522, A161701 - A161705, A000127,A161706 - A161708, A080856, A161710 - A161713, A161715, A006261. - Reinhard Zumkeller, Jun 17 2009

Cf. A063865. - Olivier Gérard, Mar 22 2010Cf. A051601. - Bruno Berselli, Aug 02 2013Cf. A077043, A002620.Sequence in context: A159243 A089140 A204555 * A129961 A133551 A114226Adjacent sequences:  A000122 A000123 A000124 * A000126 A000127 A000128

KEYWORD nonn,easy,nice

AUTHOR N. J. A. Sloane

EXTENSIONS More terms from James A. Sellers, Feb 22 2000

STATUS approved

List of OEIS sequencesFrom Wikipedia, the free encyclopedia

This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their ownWikipedia entries.

OEIS

linkName First elements Short description

A000010

Euler's totient function φ(n) 1, 1, 2, 2, 4, 2, 6, 4, 6, 4 φ(n) is the number of the positive integers not greater than n that

are prime to n

A000027

Natural number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 The natural numbers

A000032

Lucas number2, 1, 3, 4, 7, 11, 18, 29, 47, 76

L(n) = L(n − 1) + L(n − 2)

A000040

Prime number2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The prime numbers

A000045

Fibonacci number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 F(n) = F(n − 1) + F(n − 2) with F(0) = 0 and F(1) = 1

A000058

Sylvester's sequence

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443

a(n + 1) = a(n)2 − a(n) + 1, with a(0) = 2

A000073

Tribonacci number 0, 1, 1, 2, 4, 7, 13, 24, 44, 81T(n) = T(n − 1) + T(n − 2) + T(n − 3) with T(0) = 0, T(1) = T(2) = 1

A000108

Catalan number1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862  f

or n ≥ 0.

A000110

Bell number1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147

The number of partitions of a set with n elements

A00011 Euler number 1, 1, 1, 2, 5, 16, 61, 272, The number of linear extensions of the "zig-zag" poset

OEIS

linkName First elements Short description

1 1385, 7936

A000124

Lazy caterer's sequence

1, 2, 4, 7, 11, 16, 22, 29, 37, 46

The maximal number of pieces formed when slicing a pancake with ncuts

A000129

Pell number0, 1, 2, 5, 12, 29, 70, 169, 408, 985

a(0) = 0, a(1) = 1; for n > 1, a(n) = 2a(n − 1) + a(n − 2)

A000142

Factorial1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880

n! = 1·2·3·4·...·n

A000217

Triangular number0, 1, 3, 6, 10, 15, 21, 28, 36, 45

a(n) = C(n + 1, 2) = n(n + 1)/2 = 0 + 1 + 2 + ... + n

A000292

Tetrahedral number0, 1, 4, 10, 20, 35, 56, 84, 120, 165

The sum of the first n triangular numbers

A000330

Square pyramidal number

0, 1, 5, 14, 30, 55, 91, 140, 204, 285

(n(n+1)(2n+1)) / 6

The number of stacked spheres in a pyramid with a square base

A000396

Perfect number

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128

n is equal to the sum of the proper divisors of n

A000668

Mersenne prime

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111

2p − 1 if p is a prime

A007588

Stella octangula number

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, ...

Stella octangula numbers: n*(2*n2 - 1).

A000793

Landau's function 1, 1, 2, 3, 4, 6, 6, 12, 15, 20 The largest order of permutation of n elements

A000796

Decimal expansion of Pi

3, 1, 4, 1, 5, 9, 2, 6, 5, 3

A000931

Padovan sequence 1, 1, 1, 2, 2, 3, 4, 5, 7, 9 P(0) = P(1) = P(2) = 1, P(n) = P(n−2)+P(n−3)

A000945

Euclid–Mullin sequence

2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571,

a(1) = 2, a(n+1) is smallest prime factor of a(1)a(2)...a(n)+1.

OEIS

linkName First elements Short description

139

A000959

Lucky number1, 3, 7, 9, 13, 15, 21, 25, 31, 33

A natural number in a set that is filtered by a sieve

A001006

Motzkin number1, 1, 2, 4, 9, 21, 51, 127, 323, 835

The number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle

A001045

Jacobsthal number0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341

a(n) = a(n − 1) + 2a(n − 2), with a(0) = 0, a(1) = 1

A001065

sequence ofAliquot sumss(n)

0, 1, 1, 3, 1, 6, 1, 7, 4, 8 s(n) is the sum of the proper divisors of the integer n

A001113

Decimal expansion of e (mathematical constant)

2, 7, 1, 8, 2, 8, 1, 8, 2, 8

A001190

Wedderburn–Etherington number

0, 1, 1, 1, 2, 3, 6, 11, 23, 46The number of binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n − 1 nodes in all)

A001358

Semiprime4, 6, 9, 10, 14, 15, 21, 22, 25, 26

Products of two primes

A001462

Golomb sequence 1, 2, 2, 3, 3, 4, 4, 4, 5, 5 a(n) is the number of times n occurs, starting with a(1) = 1

A001608

Perrin number 3, 0, 2, 3, 2, 5, 5, 7, 10, 12 P(0) = 3, P(1) = 0, P(2) = 2; P(n) = P(n−2) + P(n−3) for n > 2

A001620

Euler–Mascheroni constant

5, 7, 7, 2, 1, 5, 6, 6, 4, 9

A001622

Decimal expansion of the golden ratio

1, 6, 1, 8, 0, 3, 3, 9, 8, 8

A002064

Cullen number1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497

n 2n + 1

A002110

Primorial1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870

The product of first n primes

A002113

Palindromic number

0, 1, 2, 3, 4, 5, 6, 7, 8, 9 A number that remains the same when its digits are reversed

A002182

Highly composite number

1, 2, 4, 6, 12, 24, 36, 48, 60, 120

A positive integer with more divisors than any smaller positive integer

A002193

Decimal expansion ofsquare root of 2

1, 4, 1, 4, 2, 1, 3, 5, 6, 2

A002201

Superior highly composite number

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720

A positive integer n for which there is an e>0 such that d(n)/ne ≥d(k)/ke for all k>1

A002378

Pronic number0, 2, 6, 12, 20, 30, 42, 56, 72, 90

n(n+1)

A00280 Composite number 4, 6, 8, 9, 10, 12, 14, 15, 16, The numbers n of the form xy for x > 1 and y > 1

OEIS

linkName First elements Short description

8 18

A002858

Ulam number 1, 2, 3, 4, 6, 8, 11, 13, 16, 18a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms; semiperfect

A002997

Carmichael number561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341

Composite numbers n such that a(n−1) == 1 (mod n) if a is prime to n

A003261

Woodall number1, 7, 23, 63, 159, 383, 895, 2047, 4607

n 2n - 1

A003459

Permutable prime2, 3, 5, 7, 11, 13, 17, 31, 37, 71

The numbers for which every permutation of digits is a prime

A005044

Alcuin's sequence0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14

number of triangles with integer sides and perimeter n

A005100

Deficient number 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 The numbers n such that σ(n) < 2n

A005101

Abundant number12, 18, 20, 24, 30, 36, 40, 42, 48, 54

The sum of divisors of n exceeds 2n

A005150

Look-and-say sequence

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211,

A = 'frequency' followed by 'digit'-indication

A005224

Aronson's sequence

1, 4, 11, 16, 24, 29, 33, 35, 39, 45

"t" is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas

A005235

Fortunate number3, 5, 7, 13, 23, 17, 19, 23, 37, 61

The smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers

A005349

Harshad numbers in base 10

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12a Harshad number in base 10 is an integer that is divisible by the sum of its digits (when written in base 10)

A005384

Sophie Germain prime

2, 3, 5, 11, 23, 29, 41, 53, 83, 89

A prime number p such that 2p+1 is also prime

A005835

Semiperfect number

6, 12, 18, 20, 24, 28, 30, 36, 40, 42

A natural number n that is equal to the sum of all or some of its proper divisors

A006037

Weird number70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792

A natural number that is abundant but not semiperfect

A006842

Farey sequencenumerators

0, 1, 0, 1, 1, 0, 1, 1, 2, 1

A006843

Farey sequencedenominators

1, 1, 1, 2, 1, 1, 3, 2, 3, 1

A006862

Euclid number2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871

1 + product of first n consecutive primes

A00688 Kaprekar number 1, 9, 45, 55, 99, 297, 703, X² = Abn + B, where 0 < B < bn X = A + B

OEIS

linkName First elements Short description

6 999, 2223, 2728

A007304

Sphenic number30, 42, 66, 70, 78, 102, 105, 110, 114, 130

Products of 3 distinct primes

A007318

Pascal's triangle 1, 1, 1, 1, 2, 1, 1, 3, 3, 1 Pascal's triangle read by rows

A007770

Happy number1, 7, 10, 13, 19, 23, 28, 31, 32, 44

The numbers whose trajectory under iteration of sum of squares of digits map includes 1

A010060

Prouhet–Thue–Morse constant

0, 1, 1, 0, 1, 0, 0, 1, 1, 0

A014080

Factorion 1, 2, 145, 40585A natural number that equals the sum of the factorials of its decimal digits

A014577

Regular paperfolding sequence

1, 1, 0, 1, 1, 0, 0, 1, 1, 1At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence

A016114

Circular prime2, 3, 5, 7, 11, 13, 17, 37, 79, 113

The numbers which remain prime under cyclic shifts of digits

A018226

Magic number (physics)

2, 8, 20, 28, 50, 82, 126A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.

A019279

Superperfect number

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056

A027641

Bernoulli number1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0

A031214

First elements in all OEISsequences

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

One of sequences referring to the OEIS itself

A033307

Decimal expansion ofChampernowne constant

1, 2, 3, 4, 5, 6, 7, 8, 9, 1 formed by concatenating the positive integers

A035513

Wythoff array 1, 2, 4, 3, 7, 6, 5, 11, 10, 9 A matrix of integers derived from the Fibonacci sequence

A036262

Gilbreath's conjecture

2, 1, 3, 1, 2, 5, 1, 0, 2, 7 Triangle of numbers arising from Gilbreath's conjecture

A037274

Home prime1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773

For n ≥ 2, a(n) = the prime that is finally reached when you start withn, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached

A046075

Undulating number101, 121, 131, 141, 151, 161, 171, 181, 191, 202

A number that has the digit form ababab

A050278

Pandigital number 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698,

Numbers containing the digits 0-9 such that each digit appears exactly once

OEIS

linkName First elements Short description

1023457869, 1023457896

A052486

Achilles number72, 108, 200, 288, 392, 432, 500, 648, 675, 800

Powerful but imperfect

A060006

Decimal expansion ofPisot–Vijayaraghavan number

1, 3, 2, 4, 7, 1, 7, 9, 5, 7 real root of x3−x−1

A076336

Sierpinski number

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909

Odd k for which   consists only of composite numbers

A076337

Riesel number509203, 762701, 777149, 790841, 992077

Odd k for which   consists only of composite numbers

A086747

Baum–Sweet sequence

1, 1, 0, 1, 1, 0, 0, 1, 0, 1a(n) = 1 if binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0

A094683

Juggler sequence 0, 1, 1, 5, 2, 11, 2, 18, 2, 27 If n mod 2 = 0 then floor(√n) else floor(n3/2)

A097942

Highly totient number

1, 2, 4, 8, 12, 24, 48, 72, 144, 240

Each number k on this list has more solutions to the equation φ(x) =k than any preceding k

A100264

Decimal expansion ofChaitin's constant

0, 0, 7, 8, 7, 4, 9, 9, 6, 9

A104272

Ramanujan prime 2, 11, 17, 29, 41, 47, 59, 67The nth Ramanujan prime is the least integer Rn for

which   ≥ n, for all x ≥ Rn.

A122045

Euler number1, 0, −1, 0, 5, 0, −61, 0, 1385, 0