## The Encyclopedia of Integer Sequences

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Citations: | 649 - 15 self |

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@MISC{Sloane_theencyclopedia,

author = {N. J. A. Sloane},

title = {The Encyclopedia of Integer Sequences},

year = {}

}

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### Abstract

This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other

### Citations

370 |
The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/ njas/sequences
- Sloane
(Show Context)
Citation Context ...if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other information. 2. To consult the database Since 1996, an electronic version =-=[20]-=- has been accessible via the Internet, at the URL http://www.research.att.com/∼njas/sequences/. If a list of numbers is entered there, the reply will display the entries for all matching sequences. Fo... |

262 |
On Numbers and Games
- Conway
- 1976
(Show Context)
Citation Context ...nts becomes sequence A7318: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, . .. Square arrays are read by antidiagonals, usually in this order: E.g. the Nim-addition table =-=[11]-=- becomes sequence A3987: a0 a2 a5 a9 . . . a1 a4 a8 . . . a3 a7 . . . a6 . . . . . . 0 1 2 3 4 . . . 1 0 3 2 5 . . . 2 3 0 1 6 . . . 3 2 1 0 7 . . . · · · 0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, ... |

148 | Gfun: a maple package for the manipulation of generating and holonomic functions in one variable
- Salvy, Zimmerman
- 1994
(Show Context)
Citation Context ...o see if the result is in the database • applying Padé approximation methods to try, for example, to express the nth term as a rational function of n (using the “gfun” package of Salvy and Zimmermann =-=[16]-=-, the “guesss” program of Derksen [5] and the “RATE” program of Kratthentaler [8]) • checking to see if changing one or two characters produces a sequence in the database. Since Superseeker carries ou... |

132 |
Groups of homotopy spheres
- Kervaire, Milnor
- 1963
(Show Context)
Citation Context ...The Poincaré conjecture. The number of differential structures on the nsphere, for n = 1, 2, . . ., 16 (A1676): 1, 1, 1?, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, as given by Kervaire and Milnor =-=[41]-=-. The Poincaré conjecture is that the third term is 1. Dedekind’s problem. The number of monotone Boolean functions of n variables, for n = 1, 2, 3, . . ., 8 (A7153): 1, 4, 18, 166, 7579, 7828352, 241... |

98 | Monstrous moonshine
- Conway, Norton
- 1979
(Show Context)
Citation Context .... ... 0, 1,1, 2,0,2, 3,3,3,3, 4,2,0,2,4, ... . Most well-defined submissions get accepted, since an open-door policy seems the best. The amazing coincidences of the Monstrous Moonshine investigations =-=[4]-=-, for example, make it difficult to say that a particular sequence, no matter how obscure, will never be of interest. Sequences that are discouraged are those that depend on an arbitrary and large par... |

34 |
Handbook of Number Theory
- Sándor, Crstici
- 2004
(Show Context)
Citation Context ... if and only if the Riemann hypothesis holds! Although the database contains a number of sequences of both of the above types, I have not made a systematic search through reference works such as [7], =-=[11]-=- and it would be nice to get many more examples. The database can also be used to save space when referring to particular sequences. When introducing the Motzkin numbers, for example, instead of givin... |

30 | On the existence of similar sublattices
- Conway, Rains, et al.
- 1999
(Show Context)
Citation Context ...This was a very useful hint in getting started on our problem. We were able to show that this is the correct condition for the A4 lattice, and to find analogous results for a number of other lattices =-=[15]-=-. (However, we have not yet found a direct connection between A4 and the quasicrystal problem. Nevertheless, the occurrence of the same numbers in the two problems cannot be entirely coincidental.) My... |

26 |
Determinantenabschätzungen für binäre Matrizen
- Ehlich
- 1964
(Show Context)
Citation Context ...rminant of an n×n {0, 1}-matrix? The values for n = 1, 2, . . .,13 are (A3432): 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, where the last two terms are due to Ehlich, and Ehlich and Zeller =-=[25]-=-, [26]. For n ≡ −1 (mod 4), Hadamard showed that the nth term is equal to (n + 1) (n+1)/2 /2 n , provided that what is now called a Hadamard matrix of order n exists. In some cases conference matrices... |

25 |
Grandes valeurs de la fonction somme des diviseurs et hypothèse de
- Robin
- 1984
(Show Context)
Citation Context ...I cannot resist mentioning sequence A57641, which gives the values of [Hn + exp(Hn)log(Hn)] − σ(n) for n ≥ 1, where Hn is the harmonic number ∑n i=1 1/i. Lagarias [9], extending earlier work of Robin =-=[15]-=-, has shown that this sequence is nonnegative if and only if the Riemann hypothesis holds! Although the database contains a number of sequences of both of the above types, I have not made a systematic... |

23 | An elementary problem equivalent to the Riemann hypothesis
- Lagarias
(Show Context)
Citation Context ...at gives a proof of your inequality. I cannot resist mentioning sequence A57641, which gives the values of [Hn + exp(Hn)log(Hn)] − σ(n) for n ≥ 1, where Hn is the harmonic number ∑n i=1 1/i. Lagarias =-=[9]-=-, extending earlier work of Robin [15], has shown that this sequence is nonnegative if and only if the Riemann hypothesis holds! Although the database contains a number of sequences of both of the abo... |

20 |
Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics
- Arnold
(Show Context)
Citation Context ...e sum of the previous entry in the same row and the entry above it in the previous row. (This is the boustrophedon or “ox-plowing” rule.) The earliest reference I have seen to this triangle is Arnold =-=[2]-=-, who calls it the Euler-Bernoulli triangle. However, it may well be much older origin. [72] gives many other references. Richard Guy [34] observed that if the entries at the beginnings of the rows ar... |

20 |
The Eternal Golden Braid. Vintage Books
- Gödel, Bach
(Show Context)
Citation Context ..., 12, 18, 26, 35,45,56, 69, 83, 98, 114, ... 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19,... are the terms not in the sequence! This is one of many fine self-generating sequences from Hofstadter =-=[38]-=-. Golomb’s sequence. The nth term is the number of times n appears (A1462): 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7,7, 7, 8, . .. The nth term is the nearest integer to (and converges to) τ ... |

14 |
Personal communication to
- Conway
(Show Context)
Citation Context ...rences of the previous row). 10 The Wythoff array This array shown in Fig. 9 has many wonderful properties, some of which are mentioned here. I learned about most of these properties from John Conway =-=[13]-=-, but this array has a long history — see Fraenkel and Kimberling [28], Kimberling [43], [44], [45], [46], [47], [48], [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of ... |

13 |
Private communication
- Knuth
- 2010
(Show Context)
Citation Context ...thms known require on the order of nCn steps, where Cn = 1 ( ) 2n n+1 n is the nth Catalan number (A108): these algorithms are due to Koehler [55] for the stamp-folding problem and to Knuth and Pratt =-=[54]-=- and Reeds [85] for the meandric problem. They are too complicated to describe here. Using these algorithms, Stephane Legendre [61] has extended the stampfolding sequence to 26 terms and the meandric ... |

12 | How to do MONTHLY problems with your computer - Nemes, Petkovsek, et al. - 1997 |

12 | Meanders: a direct enumeration approach
- Francesco, Golinelli, et al.
- 1996
(Show Context)
Citation Context ...rms and the meandric sequence to 25 terms. Knuth and Pratt [54] have computed 17 terms of the M2, M4, M6, . . . subsequence. Lando and Zvonkin [59], [60] and Di Francesco, Golinelli and Guitter [21], =-=[20]-=-, [22] have also studied these sequences. The exact rate of growth of these sequences is not known. The best bounds for M2n presently known appear to be due to Reeds, Shepp and McIlroy [85]. It is eas... |

11 |
Some canonical sequences of integers. Linear Algebra and its Applications
- Bernstein, Sloane
- 1995
(Show Context)
Citation Context ... database. If the simple lookup fails, Superseeker carries out many other tests, including: • applying over 130 transformations to the sequence, including the binomial, Euler, Möbius, etc. transforms =-=[1]-=-, and checking to see if the result is in the database • applying Padé approximation methods to try, for example, to express the nth term as a rational function of n (using the “gfun” package of Salvy... |

11 |
Finite graphs and networks
- Busacker, Saaty
- 1965
(Show Context)
Citation Context ...hydrocarbons with n atoms”, and is based on an 1875 paper of Cayley [9]. The paper is extremely hard to follow, and gives incorrect values for n = 12 and 13. The errors it contains were reproduced in =-=[7]-=-, [91] and [93], even though Herrmann [37] had pointed out these errors in 1880. As far as we can tell, a correct verion of this sequence was never published until Eric Rains and I did so in 1999 [83]... |

11 |
Folding a strip of stamps
- Koehler
- 1968
(Show Context)
Citation Context ... known that such algorithms do not exist). The best algorithms known require on the order of nCn steps, where Cn = 1 ( ) 2n n+1 n is the nth Catalan number (A108): these algorithms are due to Koehler =-=[55]-=- for the stamp-folding problem and to Knuth and Pratt [54] and Reeds [85] for the meandric problem. They are too complicated to describe here. Using these algorithms, Stephane Legendre [61] has extend... |

10 |
folding and arch statistics
- Francesco, Golinelli, et al.
- 1996
(Show Context)
Citation Context ... 26 terms and the meandric sequence to 25 terms. Knuth and Pratt [54] have computed 17 terms of the M2, M4, M6, . . . subsequence. Lando and Zvonkin [59], [60] and Di Francesco, Golinelli and Guitter =-=[21]-=-, [20], [22] have also studied these sequences. The exact rate of growth of these sequences is not known. The best bounds for M2n presently known appear to be due to Reeds, Shepp and McIlroy [85]. It ... |

10 |
Generalized Wythoff Arrays, Shuffles, and Interspersions
- Fraenkel, Kimberling
(Show Context)
Citation Context ...Fig. 9 has many wonderful properties, some of which are mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history — see Fraenkel and Kimberling =-=[28]-=-, Kimberling [43], [44], [45], [46], [47], [48], [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513). 0 1 1 2 3 5 8 13... |

10 |
Construction of Hadamard matrices of order 28
- Kimura, Ohmori
- 1986
(Show Context)
Citation Context ...terms! Enumerating Hadamard matrices. The number of Hadamard matrices of orders n = 4, 8, 12, . . ., 28 (A7299) is 1, 1, 1, 5, 3, 60, 487, where the last entry is the work of Kimura [50], [51], [52], =-=[53]-=-. The Hadamard conjecture is that such a matrix always exists if n is a multiple of 4. Judging by 6how rapidly these numbers are growing, this should not be hard to prove, yet it has remained an open... |

9 |
Skew Hadamard matrices of order 28 with Hall sets. Notes sent to
- Kimura
- 1990
(Show Context)
Citation Context ...s some more terms! Enumerating Hadamard matrices. The number of Hadamard matrices of orders n = 4, 8, 12, . . ., 28 (A7299) is 1, 1, 1, 5, 3, 60, 487, where the last entry is the work of Kimura [50], =-=[51]-=-, [52], [53]. The Hadamard conjecture is that such a matrix always exists if n is a multiple of 4. Judging by 6how rapidly these numbers are growing, this should not be hard to prove, yet it has rema... |

7 |
The combinatorics of mancalatype games
- Broline, Loeb
- 1995
(Show Context)
Citation Context ...ndreds of variants and many different names. The version to be described here is called Tchoukaillon solitaire. It has been studied by several authors (see for example Betten [5] and Broline and Loeb =-=[6]-=-). It is played on a board with a row of holes numbered 0, 1, 2, . . . (see Fig. 11). 5 4 3 2 1 0 5 4 3 2 1 0 Fig.11. A move in Tchoukaillon solitaire. 22The game begins with n stones placed anywhere... |

6 |
The branched covering CP 2 → S 4 , hyperbolicity and projective topology, Sibirsk
- Arnold
- 1988
(Show Context)
Citation Context ...1/n 2n exists and satisfies 4 ≤ µ ≤ 16. In [85] it is shown that 8.8 ≤ µ ≤ 13.01 . Besides the papers already mentioned, the meandric and stamp-folding problems have recently been discussed by Arnold =-=[1]-=-, Di Francesco [19], Harris [36], Lando and Zvonkin [59], [60], Lunnon [64], Sade [88], Smith [95] and Touchard [98]. 7 Extremal codes and lattices Let C be a binary linear self-dual code of length n ... |

5 |
Some sequences of integers
- Cameron
- 1989
(Show Context)
Citation Context ... 1, 1, 2, 6, 17, 62, 259,1230,6592, . . . We do not know what this enumerates! Many examples of similar “eigen-sequences” for other transformations of sequences can be found in Donaghey [23], Cameron =-=[8]-=-, and especially [4]. E.g. the sequence giving the number of planted achiral trees [30], [35] (A3238): 1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, ... has the property that it shifts left one place under th... |

4 | Numerical analogues of Aronson’s sequence - Cloitre, Sloane, et al. |

4 |
The number of achiral trees
- Harary, Robinson
- 1975
(Show Context)
Citation Context ... of similar “eigen-sequences” for other transformations of sequences can be found in Donaghey [23], Cameron [8], and especially [4]. E.g. the sequence giving the number of planted achiral trees [30], =-=[35]-=- (A3238): 1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, ... has the property that it shifts left one place under the “inverse Möbius transformation” given by bn = ∑ d|n ad . 12 Tchoukaillon solitaire (or Manc... |

4 |
Stolarsky interspersions, Ars Combinatoria 39
- Kimberling
- 1995
(Show Context)
Citation Context ...e of which are mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history — see Fraenkel and Kimberling [28], Kimberling [43], [44], [45], [46], =-=[47]-=-, [48], [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513). 0 1 1 2 3 5 8 13 21 34 55 1 3 4 7 11 18 29 47 76 .. 2 4 6... |

4 |
Fractal sequences and interspersions
- Kimberling
- 1997
(Show Context)
Citation Context ...re mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history — see Fraenkel and Kimberling [28], Kimberling [43], [44], [45], [46], [47], [48], =-=[49]-=-, Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513). 0 1 1 2 3 5 8 13 21 34 55 1 3 4 7 11 18 29 47 76 .. 2 4 6 10 16 26 42... |

3 |
Mathematicians get an on-line fingerprint file
- CIPRA
- 1994
(Show Context)
Citation Context ...good chance that your sequence will be there. If not, you will see a message encouraging you to submit it. Most of these applications are unspectacular, akin to looking up a word in a dictionary (cf. =-=[2]-=-). One encounters a sequence in the middle of a calculation, perhaps 1 2 4 6 10 12 16 18 22 28 30 ... , and one wants to know quickly what it is — preferably a formula for the n-th term (in this case ... |

3 |
favorite integer sequences, Sequences and their applications (Singapore
- Sloane, My
- 1998
(Show Context)
Citation Context ...1995). Disk space is cheap, and the present incarnation (excluding illustrations) contains about 72 times as much data as the 1995 book. The history of the Encyclopedia is described in more detail in =-=[19]-=-. 23. Applications Most people use the Encyclopedia to identify a sequence, as illustrated above. It has been around long enough so that there is a good chance that your sequence will be there. If no... |

3 |
Binomial self-inverse sequences and tangent coefficients
- Donaghey
- 1976
(Show Context)
Citation Context ...A661) is 1, 0, 1, 1, 2, 6, 17, 62, 259,1230,6592, . . . We do not know what this enumerates! Many examples of similar “eigen-sequences” for other transformations of sequences can be found in Donaghey =-=[23]-=-, Cameron [8], and especially [4]. E.g. the sequence giving the number of planted achiral trees [30], [35] (A3238): 1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, ... has the property that it shifts left one p... |

3 |
Monthly 82
- Math
- 1975
(Show Context)
Citation Context ...ments that include the initial term of the current permutation, how many reversals are needed to transform an arbitrary permutation of n letters to the identity permutation? To state this another way =-=[24]-=-: The chef in our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (s... |

3 |
Problem 5407
- Golomb
- 1966
(Show Context)
Citation Context ...s the number of times n appears (A1462): 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7,7, 7, 8, . .. The nth term is the nearest integer to (and converges to) τ 2−τ n τ −1 , where τ = (1 + √ 5)/2 =-=[31]-=-, [33, Section E25]. Wilson’s primeth recurrence. an+1 is the an-th prime (A7097), shown in Fig. 2. The sequence was sent in by R. G. Wilson V [102], and the last few terms were computed by P. Zimmerm... |

3 |
Hadamard matrices of order 28 with automorphism groups of order two
- Kimura
(Show Context)
Citation Context ...well as some more terms! Enumerating Hadamard matrices. The number of Hadamard matrices of orders n = 4, 8, 12, . . ., 28 (A7299) is 1, 1, 1, 5, 3, 60, 487, where the last entry is the work of Kimura =-=[50]-=-, [51], [52], [53]. The Hadamard conjecture is that such a matrix always exists if n is a multiple of 4. Judging by 6how rapidly these numbers are growing, this should not be hard to prove, yet it ha... |

3 |
Classification of Hadamard matrices of order 28, Discrete Math
- Kimura
- 1994
(Show Context)
Citation Context ... more terms! Enumerating Hadamard matrices. The number of Hadamard matrices of orders n = 4, 8, 12, . . ., 28 (A7299) is 1, 1, 1, 5, 3, 60, 487, where the last entry is the work of Kimura [50], [51], =-=[52]-=-, [53]. The Hadamard conjecture is that such a matrix always exists if n is a multiple of 4. Judging by 6how rapidly these numbers are growing, this should not be hard to prove, yet it has remained a... |

2 |
RATE-A Mathematica guessing machine, available electronically from http://euler.univ-lyon1.fr/home/kratt/rate/rate.html
- Krattenthaler
(Show Context)
Citation Context ...y, for example, to express the nth term as a rational function of n (using the “gfun” package of Salvy and Zimmermann [16], the “guesss” program of Derksen [5] and the “RATE” program of Kratthentaler =-=[8]-=-) • checking to see if changing one or two characters produces a sequence in the database. Since Superseeker carries out a nontrivial amount of calculation, users are asked to submit only one sequence... |

2 |
combinatorische Probleme [Four combinatorial problems
- Schröder, Vier
(Show Context)
Citation Context ... A1003) arising from “Schröder’s second problem”, and are also known as “super-Catalan numbers”. The reply (an abridged version is shown in Figure 1) gives 21 references, ranging from Schröder (1870) =-=[17]-=- to articles published electronically in the last few years. There is an ∗ Neil J. A. Sloane is with AT&T Shannon Labs, Florham Park, NJ. His email address is njas@research.att.com.explicit formula: ... |

2 |
Ueber die analytischen Figuren, welche in der Mathematik B€aume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem
- Cayley
(Show Context)
Citation Context ...maintain the dignity of the database, and partly because A22 was only known to 11 terms! Sequence A22 gives the number of “centered hydrocarbons with n atoms”, and is based on an 1875 paper of Cayley =-=[9]-=-. The paper is extremely hard to follow, and gives incorrect values for n = 12 and 13. The errors it contains were reproduced in [7], [91] and [93], even though Herrmann [37] had pointed out these err... |

2 |
Some crazy sequences, videotaped talk at
- Conway
(Show Context)
Citation Context ...428095424619 75063692618249 2586559730396077 Fig.2. an+1 is the an-th prime. These were sent in by B. Recamán [84]. How fast do they grow? The $10,000 sequence. In a colloquium talk at AT&T Bell Labs =-=[12]-=-, John Conway discussed the sequence (A4001) defined by (for n ≥ 3) 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, . . . a(n + 1) = a(a(n)) + a(n + 1 − a(n)) . (In words, a(n + 1) is the a(n)th te... |

2 |
On a sequence generated by a sieving process, Riveon Lematematika 11
- David
- 1957
(Show Context)
Citation Context ... . . are (sequence A2491): 1, 2, 4, 6, 10, 12, 18, 22, 30, 34, 42, . . . This sequence has some very nice properties. It has been investigated by (in addition to the references mentioned above) David =-=[17]-=-, Erdős and Jabotinsky [27] and Smarandache [94]. (i) t(n) can be obtained by starting with n and successively rounding up to the next multiple of n − 1, n − 2, . . . , 2, 1. E.g. if n = 10, we obtain... |

2 |
Orderings of the Set of All Positive Fibonacci Sequences
- Kimberling
- 1993
(Show Context)
Citation Context ...onderful properties, some of which are mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history — see Fraenkel and Kimberling [28], Kimberling =-=[43]-=-, [44], [45], [46], [47], [48], [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513). 0 1 1 2 3 5 8 13 21 34 55 1 3 4 7... |

2 |
The Zeckendorf Array Equals the Wythoff Array." The Fibonacci Quarterly 23A
- Kimberling
- 1995
(Show Context)
Citation Context ...hich are mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history — see Fraenkel and Kimberling [28], Kimberling [43], [44], [45], [46], [47], =-=[48]-=-, [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513). 0 1 1 2 3 5 8 13 21 34 55 1 3 4 7 11 18 29 47 76 .. 2 4 6 10 16... |

1 |
An algorithm to compute generalized Padé-Hermite forms
- Derken
- 1994
(Show Context)
Citation Context ... • applying Padé approximation methods to try, for example, to express the nth term as a rational function of n (using the “gfun” package of Salvy and Zimmermann [16], the “guesss” program of Derksen =-=[5]-=- and the “RATE” program of Kratthentaler [8]) • checking to see if changing one or two characters produces a sequence in the database. Since Superseeker carries out a nontrivial amount of calculation,... |

1 |
Geest, Home primes < 100 and beyond, published electronically at www.worldofnumbers.com/topic1.htm
- De
- 2003
(Show Context)
Citation Context ...e places where volunteers can help. One of the pleasures of maintaining the database is seeing the endless flow of new sequences. I will end by mentioning a few recent examples: Home primes (A37274), =-=[6]-=-: a(n) is the prime reached when you start with n, concatenate its prime factors, and repeat until a prime is reached (a(n) is defined to be −1 if no prime is ever reached, although it is conjectured ... |

1 |
What is the best way to lace your shoes?, Nature
- Polster
(Show Context)
Citation Context ...n ≥ 3, a(n) is the smallest natural number not in {a(k) : 1 ≤ k ≤ n − 1} with the property that gcd{a(n − 1),a(n)} ≥ 2: 1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 .... Lacing a shoe (A78601), =-=[14]-=-: Number of ways to lace a shoe that has n pairs of eyelets. The lace must follow a Hamiltonian path through the 2n eyelets, and at least one of the neighbors of every eyelet must be on the other side... |

1 |
Kalahari and the sequence “Sloane No
- Betten
- 1988
(Show Context)
Citation Context ...ient board games, with hundreds of variants and many different names. The version to be described here is called Tchoukaillon solitaire. It has been studied by several authors (see for example Betten =-=[5]-=- and Broline and Loeb [6]). It is played on a board with a row of holes numbered 0, 1, 2, . . . (see Fig. 11). 5 4 3 2 1 0 5 4 3 2 1 0 Fig.11. A move in Tchoukaillon solitaire. 22The game begins with... |

1 |
Computation of large values of π(x), preprint
- Deléglise
- 1996
(Show Context)
Citation Context ...’s primeth recurrence. an+1 is the an-th prime (A7097), shown in Fig. 2. The sequence was sent in by R. G. Wilson V [102], and the last few terms were computed by P. Zimmermann [103] and M. Deléglise =-=[18]-=-. Their algorithm is a slightly speeded up version of an algorithm for computing π(x), the number of primes not exceeding x, due to J. C. Lagarias, V. S. Miller and A. M. Odlyzko (see [57]). It is qui... |

1 |
On a sequence of integers generated by a sieving process
- Erdős, Jabotinsky
- 1958
(Show Context)
Citation Context ...1, 2, 4, 6, 10, 12, 18, 22, 30, 34, 42, . . . This sequence has some very nice properties. It has been investigated by (in addition to the references mentioned above) David [17], Erdős and Jabotinsky =-=[27]-=- and Smarandache [94]. (i) t(n) can be obtained by starting with n and successively rounding up to the next multiple of n − 1, n − 2, . . . , 2, 1. E.g. if n = 10, we obtain 10 → 18 → 24 → 28 → 30 → 3... |