Sylvester's sequence

Formally, Sylvester's sequence can be defined by the formula{{cite OEIS|A000058| Sylvester's sequence}}

:$s\_n\; =\; 1\; +\; \backslash prod\_\{i=0\}^\{n-1\}\; s\_i.$

The product of the empty set is 1,{{sfnp|Nešetřil|Matoušek|1998}} so this formula gives s₀ = 2, without need of a separate base case.

Alternatively, one may define the sequence by the recurrence{{sfnp|Vardi|1991}}

:$\backslash displaystyle\; s\_i\; =\; s\_\{i-1\}(s\_\{i-1\}-1)+1,$ with the base case s₀ = 2.

It is straightforward to show by induction that this is equivalent to the other definition.A proof by induction is given by {{harvtxt|Sylvester|1880}}, p. 333.

The Sylvester numbers grow doubly exponentially as a function of n. Specifically, it can be shown that

:$s\_n\; =\; \backslash left\backslash lfloor\; E^\{2^\{n+1\}\}+\backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor,\backslash !$

for a number E that is approximately 1.26408473530530...{{harvtxt|Graham|Knuth|Patashnik|1989}}, formula 4.17, p. 109, and exercise 4.37, p. 147; see also {{harvtxt|Golomb|1963}}. {{OEIS|A076393}}. This formula has the effect of the following algorithm:

:s₀ is the nearest integer to E^{{hairsp}}2; s₁ is the nearest integer to E^{{hairsp}}4; s₂ is the nearest integer to E^{{hairsp}}8; for s_n, take E^{{hairsp}}2, square it n more times, and take the nearest integer.

This would only be a practical algorithm if we had a better way of calculating E to the requisite number of places than calculating s_n and taking its repeated square root.{{sfnp|Graham|Knuth|Patashnik|1989|p=109}}

The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of Fermat numbers F_n{{hairsp}}; the Fermat numbers are usually defined by a doubly exponential formula, $2^\{2^n\}\; \backslash !+\; 1$, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:{{cite OEIS|A000215|Fermat numbers}}

:$F\_n\; =\; 2\; +\; \backslash prod\_\{i=0\}^\{n-1\}\; F\_i.$

The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series:This series is the starting point for {{harvtxt|Sylvester|1880}}

:$\backslash sum\_\{i=0\}^\{\backslash infty\}\; \backslash frac\{1\}\{s\_i\}\; =\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{7\}\; +\; \backslash frac\{1\}\{43\}\; +\; \backslash frac\{1\}\{1807\}\; +\; \backslash cdots.$

The partial sums of this series have a simple form,

:$\backslash sum\_\{i=0\}^\{j-1\}\; \backslash frac1\{s\_i\}\; =\; 1\; -\; \backslash frac\{1\}\{s\_j-1\}\; =\; \backslash frac\{s\_j-2\}\{s\_j-1\},$

which is already in lowest terms.{{sfnp|Sylvester|1880|p=334}} This may be proved by induction, or more directly by noting that the recursion implies that

:$\backslash frac\{1\}\{s\_i-1\}-\backslash frac\{1\}\{s\_\{i+1\}-1\}\; =\; \backslash frac\{1\}\{s\_i\},$

so the sum telescopes{{sfnp|Sylvester|1880|p=334}}

:$\backslash sum\_\{i=0\}^\{j-1\}\; \backslash frac\{1\}\{s\_i\}\; =\; \backslash sum\_\{i=0\}^\{j-1\}\; \backslash left(\; \backslash frac\{1\}\{s\_i-1\}-\backslash frac\{1\}\{s\_\{i+1\}-1\}\; \backslash right)\; =\; \backslash frac\{1\}\{s\_0-1\}\; -\; \backslash frac\{1\}\{s\_j-1\}\; =\; 1\; -\; \backslash frac\{1\}\{s\_j-1\}.$

Since this sequence of partial sums (s_j − 2)/(s_j − 1) converges to one, the overall series forms an infinite Egyptian fraction representation of the number one:

:$1\; =\; \backslash frac12\; +\; \backslash frac13\; +\; \backslash frac17\; +\; \backslash frac1\{43\}\; +\; \backslash frac1\{1807\}\; +\; \backslash cdots.$

One can find finite Egyptian fraction representations of one, of any length, by truncating this series and subtracting one from the last denominator:

:$1\; =\; \backslash tfrac12\; +\; \backslash tfrac13\; +\; \backslash tfrac16,\; \backslash quad\; 1\; =\; \backslash tfrac12\; +\; \backslash tfrac13\; +\; \backslash tfrac17\; +\; \backslash tfrac1\{42\},\; \backslash quad\; 1\; =\; \backslash tfrac12\; +\; \backslash tfrac13\; +\; \backslash tfrac17\; +\; \backslash tfrac1\{43\}\; +\; \backslash tfrac1\{1806\},\backslash quad\; \backslash dots.$

The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any k-term Egyptian fraction.This claim is commonly attributed to {{harvtxt|Curtiss|1922}}, but {{harvtxt|Miller|1919}} appears to be making the same statement in an earlier paper. See also {{harvtxt|Rosenman|Underwood|1933}}, {{harvtxt|Salzer|1947}}, {{harvtxt|Soundararajan|2005}}, and {{harvtxt|Nathanson|2023}}. For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806, 1) requires at least five terms.

It is possible to interpret the Sylvester sequence as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one.{{sfnp|Nathanson|2023}}

As Sylvester himself observed, Sylvester's sequence seems to be unique in having such quickly growing values, while simultaneously having a series of reciprocals that converges to a rational number. This sequence provides an example showing that double-exponential growth is not enough to cause an integer sequence to be an irrationality sequence.{{sfnp|Guy|2004}}

To make this more precise, it follows from results of {{harvtxt|Badea|1993}} that, if a sequence of integers $a\_n$ grows quickly enough that

:$a\_n\backslash ge\; a\_\{n-1\}^2-a\_\{n-1\}+1,$

and if the series

:$A=\backslash sum\backslash frac1\{a\_i\}$

converges to a rational number A, then, for all n after some point, this sequence must be defined by the same recurrence

:$a\_n=\; a\_\{n-1\}^2-a\_\{n-1\}+1$

that can be used to define Sylvester's sequence.{{sfnp|Badea|1993}}

{{harvtxt|Erdős|Graham|1980}} conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,{{sfnp|Erdős|Graham|1980}}

:$\backslash lim\_\{n\backslash rightarrow\backslash infty\}\; \backslash frac\{a\_n\}\{a\_\{n-1\}^2\}=1.$

{{harvtxt|Badea|1995}} surveys progress related to this conjecture; see also {{harvtxt|Brown|1979}}.{{sfnmp

| 1a1 = Badea | 1y = 1995

| 2a1 = Brown | 2y = 1979

}}

If i < j, it follows from the definition that s_j ≡ 1 (mod s_i{{hairsp}}). Therefore, every two numbers in Sylvester's sequence are relatively prime. The sequence can be used to prove that there are infinitely many prime numbers, as any prime can divide at most one number in the sequence. More strongly, no prime factor of a number in the sequence can be congruent to 5 modulo 6, and the sequence can be used to prove that there are infinitely many primes congruent to 7 modulo 12.{{sfnp|Guy|Nowakowski|1975}} No term can be a perfect power.{{sfnp|Makki Naciri|2024}}

{{unsolved|mathematics|Are all the terms in Sylvester's sequence squarefree?}}

Much remains unknown about the factorization of the numbers in Sylvester's sequence. For instance, it is not known if all numbers in the sequence are squarefree, although all the known terms are.{{harvtxt|Graham|Knuth|Patashnik|1989}}, research problem 4.65, p. 151; {{harvtxt|Vardi|1991}}; see also {{harvtxt|Chentouf|2020}}

As {{harvtxt|Vardi|1991}} describes, it is easy to determine which Sylvester number (if any) a given prime p divides: simply compute the recurrence defining the numbers modulo p until finding either a number that is congruent to zero (mod p) or finding a repeated modulus.{{sfnp|Vardi|1991}} Using this technique he found that 1166 out of the first three million primes are divisors of Sylvester numbers,This appears to be a typo, as Andersen finds 1167 prime divisors in this range. and that none of these primes has a square that divides a Sylvester number.

The set of primes that can occur as factors of Sylvester numbers is of density zero in the set of all primes:{{sfnp|Jones|2006}} indeed, the number of such primes less than x is $O(\backslash pi(x)\; /\; \backslash log\backslash log\backslash log\; x)$.{{sfnp|Odoni|1985}}

The following table shows known factorizations of these numbers (except s₀ ... s₃, which are all prime):All prime factors p of Sylvester numbers s_n with p < 5{{e|7}} and n ≤ 200 are listed by Vardi. Ken Takusagawa lists the [http://kenta.blogspot.com/2006/01/factoring-sylvesters-sequence.html factorizations up to s₉] and the [http://kenta.blogspot.com/2006/04/sylvester-10th-factored.html factorization of s₁₀]. The remaining factorizations are from [http://primerecords.dk/sylvester-factors.htm a list of factorizations of Sylvester's sequence] maintained by Jens Kruse Andersen. Retrieved 2014-06-13.

As is customary, Pn and Cn denote prime numbers and unfactored composite numbers n digits long.

{{harvtxt|Boyer|Galicki|Kollár|2005}} use the properties of Sylvester's sequence to define large numbers of Sasakian Einstein manifolds having the differential topology of odd-dimensional spheres or exotic spheres. They show that the number of distinct Sasakian Einstein metrics on a topological sphere of dimension 2n  − 1 is at least proportional to s_n and hence has double exponential growth with n.{{sfnp|Boyer|Galicki|Kollár|2005}}

As {{harvtxt|Galambos|Woeginger|1995}} describe, {{harvtxt|Brown|1979}} and {{harvtxt|Liang|1980}} used values derived from Sylvester's sequence to construct lower bound examples for online bin packing algorithms.{{sfnmp

| 1a1 = Galambos | 1a2 = Woeginger | 1y = 1995

| 2a1 = Brown | 2y = 1979

| 3a1 = Liang | 3y = 1980

}}

{{harvtxt|Seiden|Woeginger|2005}} similarly use the sequence to lower bound the performance of a two-dimensional cutting stock algorithm.In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of {{harvtxt|Salzer|1947}} on closest approximation.

Znám's problem concerns sets of numbers such that each number in the set divides but is not equal to the product of all the other numbers, plus one. Without the inequality requirement, the values in Sylvester's sequence would solve the problem; with that requirement, it has other solutions derived from recurrences similar to the one defining Sylvester's sequence. Solutions to Znám's problem have applications to the classification of surface singularities (Brenton and Hill 1988) and to the theory of nondeterministic finite automata.{{sfnp|Domaratzki|Ellul|Shallit|Wang|2005}}

{{harvtxt|Curtiss|1922}} describes an application of the closest approximations to one by k-term sums of unit fractions, in lower-bounding the number of divisors of any perfect number, and {{harvtxt|Miller|1919}} uses the same property to upper bound the size of certain groups.{{sfnmp

| 1a1 = Curtiss | 1y = 1922

| 2a1 = Miller | 2y = 1919

}}

• Cahen's constant
• Primary pseudoperfect number
• Leonardo number

{{reflist|30em}}

{{refbegin|30em}}

• {{cite journal

| last = Badea | first = Catalin

| title = A theorem on irrationality of infinite series and applications

| year = 1993

| journal = Acta Arithmetica

| volume = 63

| issue = 4

| pages = 313–323

| mr = 1218459

| doi = 10.4064/aa-63-4-313-323

| doi-access = free

}}

• {{cite web

|last = Badea

|first = Catalin

|title = On some criteria for irrationality for series of positive rationals: a survey

|year = 1995

|url = http://www.lacim.uqam.ca/~plouffe/OEIS/citations/caen.pdf

|url-status = dead

|archive-url = https://web.archive.org/web/20080911223925/http://www.lacim.uqam.ca/~plouffe/OEIS/citations/caen.pdf

|archive-date = 2008-09-11

}}

• {{cite journal

| last1 = Boyer | first1 = Charles P.

| last2 = Galicki | first2 = Krzysztof

| last3 = Kollár | first3 = János

| title = Einstein metrics on spheres

| journal = Annals of Mathematics

| volume = 162

| issue = 1

| year = 2005

| pages = 557–580

| arxiv = math.DG/0309408 | mr = 2178969

| doi = 10.4007/annals.2005.162.557

| s2cid = 13945306

}}

• {{cite journal

| last1 = Brenton | first1 = Lawrence | last2 = Hill | first2 = Richard

| title = On the Diophantine equation 1=Σ1/n_i + 1/Πn_i and a class of homologically trivial complex surface singularities

| journal = Pacific Journal of Mathematics

| url = http://projecteuclid.org/euclid.pjm/1102689567

| volume = 133

| issue = 1

| year = 1988

| pages = 41–67

| mr = 0936356 | doi=10.2140/pjm.1988.133.41

| doi-access = free

}}

• {{cite book

| last = Brown | first = D. J.

| title = A lower bound for on-line one-dimensional bin packing algorithms

| series = Tech. Rep. R-864

| year = 1979

| publisher = Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign

}}

• {{cite journal

| last = Chentouf | first = A. Anas

| issue = 2

| journal = Parabola

| title = On Sylvester's Sequence and some of its properties

| url = https://parabola.unsw.edu.au/files/articles/2020-2029/volume-56-2020/issue-2/vol56_no2_2.pdf

| volume = 56

| year = 2020}}

• {{cite journal

| last = Curtiss | first = D. R. | authorlink = David Raymond Curtiss| title = On Kellogg's diophantine problem

| journal = American Mathematical Monthly

| jstor = 2299023

| volume = 29

| year = 1922

| issue = 10

| pages = 380–387

| doi = 10.2307/2299023

}}

• {{cite journal

| last1 = Domaratzki | first1 = Michael

| last2 = Ellul | first2 = Keith

| author3-link = Jeffrey Shallit | last3 = Shallit | first3 = Jeffrey

| last4 = Wang | first4 = Ming-Wei

| title = Non-uniqueness and radius of cyclic unary NFAs

| journal = International Journal of Foundations of Computer Science

| volume = 16

| issue = 5

| pages = 883–896

| year = 2005

| url = http://www.cs.umanitoba.ca/~mdomarat/pubs/DESW_dcfs.ps

| mr = 2174328

| doi = 10.1142/S0129054105003352

}}

• {{cite book

| author1-link = Paul Erdős | last1 = Erdős | first1 = Paul

| author2-link = Ronald Graham | last2 = Graham | first2 = Ronald L.

| year = 1980

| title = Old and new problems and results in combinatorial number theory

| publisher = Monographies de L'Enseignement Mathématique, No. 28, Univ. de Genève

| mr = 0592420

}}

• {{cite journal

| last1 = Galambos | first1 = Gábor | last2 = Woeginger | first2 = Gerhard J. | author2-link = Gerhard J. Woeginger

| title = On-line bin packing — A restricted survey

| journal = Mathematical Methods of Operations Research

| volume = 42

| issue = 1

| year = 1995

| doi = 10.1007/BF01415672

| mr = 1346486

| pages = 25

| s2cid = 26692460 | doi-access = free

}}

• {{cite journal

| last1 = Golomb | first1 = Solomon W.

| authorlink = Solomon Golomb

| title = On certain nonlinear recurring sequences

| year = 1963

| journal = American Mathematical Monthly

| volume = 70

| issue = 4

| pages = 403–405

| jstor = 2311857

| mr = 0148605

| doi = 10.2307/2311857

}}

• {{cite book

| last1 = Graham | first1 = Ronald | author1-link = Ronald Graham

| last2 = Knuth | first2 = Donald E. | author2-link = Donald Knuth

| last3 = Patashnik | first3 = Oren | author3-link = Oren Patashnik

| edition = 2e

| isbn = 978-0-201-55802-9

| publisher = Addison-Wesley

| title = Concrete Mathematics

| title-link = Concrete Mathematics

| year = 1989}}

• {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved Problems in Number Theory | publisher=Springer-Verlag |edition=3rd | year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | contribution=E24 Irrationality sequences|page=346|url=https://books.google.com/books?id=1AP2CEGxTkgC&pg=PA346 }}
• {{cite journal

| last1 = Guy | first1 = Richard

| last2 = Nowakowski | first2 = Richard

| issue = 2

| journal = Delta (Waukesha)

| mr = 0384675

| pages = 49–63

| title = Discovering primes with Euclid

| volume = 5

| year = 1975

}}

• {{cite journal | last = Jones | first = Rafe | title = The density of prime divisors in the arithmetic dynamics of quadratic polynomials

| year = 2006

| arxiv = math.NT/0612415| doi=10.1112/jlms/jdn034 | volume=78 | issue = 2 | journal=Journal of the London Mathematical Society | pages=523–544| bibcode = 2006math.....12415J | s2cid = 15310955 }}

• {{cite journal

| last = Liang | first = Frank M.

| title = A lower bound for on-line bin packing

| year = 1980

| journal = Information Processing Letters

| volume = 10

| issue = 2

| pages = 76–79

| mr = 0564503

| doi = 10.1016/S0020-0190(80)90077-0

}}

• {{cite journal

| last = Makki Naciri | first = Abderrahim

| journal = Integers

| mr = 4843289

| article-number = A117

| pages = 1–7

| title = A note on Sylvester's sequence

| url = https://math.colgate.edu/~integers/y117/y117.pdf

| volume = 24

| year = 2024}}

• {{cite book

| last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil

| last2 = Matoušek | first2 = Jiří | author2-link = Jiří Matoušek (mathematician)

| isbn = 0-19-850207-9

| page = 12

| publisher = Oxford University Press

| title = Invitation to Discrete Mathematics

| year = 1998}}

• {{cite journal

| last = Miller | first = G. A.

| title = Groups possessing a small number of sets of conjugate operators

| journal = Transactions of the American Mathematical Society

| year = 1919

| volume = 20

| issue = 3

| pages = 260–270

| jstor = 1988867

| doi = 10.2307/1988867

| doi-access = free

}}

• {{cite journal

| last = Nathanson | first = Melvyn B.

| date = January 2023

| doi = 10.1016/j.jnt.2022.07.005

| journal = Journal of Number Theory

| pages = 208–234

| title = Underapproximation by Egyptian fractions

| volume = 242| arxiv = 2202.00191

}}

• {{cite journal | last=Odoni | first=R. W. K. | title=On the prime divisors of the sequence w_n+1 =1+w₁⋯w_n | zbl=0574.10020 | journal=Journal of the London Mathematical Society |series=Series II | volume=32 | pages=1–11 | year=1985| doi=10.1112/jlms/s2-32.1.1 }}
• {{cite journal

| last1 = Rosenman | first1 = Martin

| last2 = Underwood | first2 = F.

| title = Problem 3536

| journal = American Mathematical Monthly

| volume = 40

| issue = 3

| year = 1933

| jstor = 2301036

| pages = 180–181

| doi = 10.2307/2301036

}}

• {{cite journal

| last = Salzer | first = H. E.

| journal = American Mathematical Monthly

| year = 1947

| title = The approximation of numbers as sums of reciprocals

| volume = 54

| issue = 3

| pages = 135–142

| jstor = 2305906

| mr = 0020339

| doi = 10.2307/2305906

}}

• {{cite journal

| last1 = Seiden | first1 = Steven S. | last2 = Woeginger | first2 = Gerhard J. | author2-link = Gerhard J. Woeginger

| title = The two-dimensional cutting stock problem revisited

| journal = Mathematical Programming

| volume = 102

| issue = 3

| year = 2005

| pages = 519–530

| doi = 10.1007/s10107-004-0548-1

| mr = 2136225

| s2cid = 35815524 }}

• {{cite arXiv

| last = Soundararajan | first = K.

| year = 2005

| title = Approximating 1 from below using n Egyptian fractions

| eprint = math.CA/0502247

}}

• {{cite journal

| last = Sylvester | first = J. J.

| authorlink = J. J. Sylvester

| title = On a point in the theory of vulgar fractions

| jstor = 2369261

| journal = American Journal of Mathematics

| volume = 3

| issue = 4

| year = 1880

| pages = 332–335

| doi = 10.2307/2369261

}}

• {{cite book

| last = Vardi | first = Ilan

| title = Computational Recreations in Mathematica

| year = 1991

| publisher = Addison-Wesley

| pages = 82–89

| isbn = 0-201-52989-0

}}

{{refend}}

• [https://web.archive.org/web/20000516205029/http://mathpages.com/home/kmath455.htm Irrationality of Quadratic Sums], from K. S. Brown's MathPages.
• {{mathworld | title = Sylvester's Sequence | urlname = SylvestersSequence}}

{{good article}}

Category:Egyptian fractions

Category:Integer sequences

Category:Series (mathematics)

Category:Number theory

Category:Recurrence relations