A007996 -id:A007996 - cp's OEIS frontend

A180871 Index of term in Sylvester's sequence A000058 divisible by prime A007996(n).

0, 1, 2, 4, 3, 11, 4, 9, 6, 6, 6, 29, 64, 42, 9, 59, 10, 80, 39, 103, 140, 41, 137, 53, 69, 146, 104, 14, 92, 15, 117, 199, 75, 98, 316, 233, 28, 92, 281, 44, 136, 26, 258, 7, 38, 6, 176, 126, 74, 59, 89, 61, 45, 79, 13, 448, 119, 180, 290, 184, 348, 502, 508, 161, 7, 265, 229

Offset: 1

T. D. Noe, Sep 25 2010

nonn

Because all terms of Sylvester's sequence are coprime to each other, each prime in A007996 divides only one term of A000058. The Mathematica program computes both the primes in A007996 and the terms in this sequence. Using modular arithmetic, it is easy to see that if prime p divides A000058(k) for some k, then we must have k < p. In practice, k < 5*sqrt(p).

An open problem is to prove that all terms of Sylvester's sequence are squarefree or to find a counterexample. Using the p from A007996 and k found here, it is simple to determine whether A000058(k) = 0 (mod p^2). No p < 10^10 was found to have this property.

			A000058(4) = 1807 = 43 * 181 = A007996(4) * A007996(7), so a(4) = a(7) = 4. - _Jonathan Sondow_, Jan 26 2014

Max Alekseyev, Table of n, a(n) for n = 1..12046 (first 8181 terms are also given at the Andersen link)
Jens Kruse Andersen, Factorization of Sylvester's sequence

Cf. A000058, A007996, A126263.

t={}; p=1; While[Length[t]<100, p=NextPrime[p]; s=Mod[2,p]; k=0; modSet={}; While[s>0 && !MemberQ[modSet,s], AppendTo[modSet,s]; k++; s=Mod[s^2-s+1,p]]; If[s==0, AppendTo[t,{p,k}]]]; Transpose[t][[2]]

A000058(a(n)) == 0 (mod A007996(n)) implies a(n) < A007996(n). - Jonathan Sondow, Jan 26 2014

Definition clarified by Jonathan Sondow, Jan 26 2014

A096263 Duplicate of A007996.

Original entry on oeis.org

2, 3, 7, 13, 43, 73, 139, 181, 547, 607, 1033, 1171, 1459, 1861, 1987, 2029, 2287

Offset: 1

dead

A000058 Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.

Original entry on oeis.org

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807

Offset: 0

N. J. A. Sloane

Also called Euclid numbers, because a(n) = a(0)*a(1)*...*a(n-1) + 1 for n>0, with a(0)=2. - Jonathan Sondow, Jan 26 2014
Another version of this sequence is given by A129871, which starts with 1, 2, 3, 7, 43, 1807, ... .
The greedy Egyptian representation of 1 is 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... .
Take a square. Divide it into 2 equal rectangles by drawing a horizontal line. Divide the upper rectangle into 2 squares. Now you can divide the lower one into another 2 squares, but instead of doing so draw a horizontal line below the first one so you obtain a (2+1 = 3) X 1 rectangle which can be divided in 3 squares. Now you have a 6 X 1 rectangle at the bottom. Instead of dividing it into 6 squares, draw another horizontal line so you obtain a (6+1 = 7) X 1 rectangle and a 42 X 1 rectangle left, etc. - Néstor Romeral Andrés, Oct 29 2001
More generally one may define f(1) = x_1, f(2) = x_2, ..., f(k) = x_k, f(n) = f(1)*...*f(n-1)+1 for n > k and natural numbers x_i (i = 1, ..., k) which satisfy gcd(x_i, x_j) = 1 for i <> j. By definition of the sequence we have that for each pair of numbers x, y from the sequence gcd(x, y) = 1. An interesting property of a(n) is that for n >= 2, 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) = (a(n)-2)/(a(n)-1). Thus we can also write a(n) = (1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) - 2 )/( 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) - 1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), May 10 2001; [corrected by Michel Marcus, Mar 27 2019]
A greedy sequence: a(n+1) is the smallest integer > a(n) such that 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n+1) doesn't exceed 1. The sequence gives infinitely many ways of writing 1 as the sum of Egyptian fractions: Cut the sequence anywhere and decrement the last element. 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/3 + 1/7 + 1/42 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = ... . - Ulrich Schimke, Nov 17 2002; [corrected by Michel Marcus, Mar 27 2019]
Consider the mapping f(a/b) = (a^3 + b)/(a + b^3). Starting with a = 1, b = 2 and carrying out this mapping repeatedly on each new (reduced) rational number gives 1/2, 1/3, 4/28 = 1/7, 8/344 = 1/43, ..., i.e., 1/2, 1/3, 1/7, 1/43, 1/1807, ... . Sequence contains the denominators. Also the sum of the series converges to 1. - Amarnath Murthy, Mar 22 2003
a(1) = 2, then the smallest number == 1 (mod all previous terms). a(2n+6) == 443 (mod 1000) and a(2n+7) == 807 (mod 1000). - Amarnath Murthy, Sep 24 2003
An infinite coprime sequence defined by recursion.
Apart from the initial 2, a subsequence of A002061. It follows that no term is a square.
It appears that a(k)^2 + 1 divides a(k+1)^2 + 1. - David W. Wilson, May 30 2004. This is true since a(k+1)^2 + 1 = (a(k)^2 - a(k) + 1)^2 +1 = (a(k)^2-2*a(k)+2)*(a(k)^2 + 1) (a(k+1)=a(k)^2-a(k)+1 by definition). - Pab Ter (pabrlos(AT)yahoo.com), May 31 2004
In general, for any m > 0 coprime to a(0), the sequence a(n+1) = a(n)^2 - m*a(n) + m is infinite coprime (Mohanty). This sequence has (m,a(0))=(1,2); (2,3) is A000215; (1,4) is A082732; (3,4) is A000289; (4,5) is A000324.
Any prime factor of a(n) has -3 as its quadratic residue (Granville, exercise 1.2.3c in Pollack).
Note that values need not be prime, the first composites being 1807 = 13 * 139 and 10650056950807 = 547 * 19569939581. - Jonathan Vos Post, Aug 03 2008
If one takes any subset of the sequence comprising the reciprocals of the first n terms, with the condition that the first term is negated, then this subset has the property that the sum of its elements equals the product of its elements. Thus -1/2 = -1/2, -1/2 + 1/3 = -1/2 * 1/3, -1/2 + 1/3 + 1/7 = -1/2 * 1/3 * 1/7, -1/2 + 1/3 + 1/7 + 1/43 = -1/2 * 1/3 * 1/7 * 1/43, and so on. - Nick McClendon, May 14 2009
(a(n) + a(n+1)) divides a(n)*a(n+1)-1 because a(n)*a(n+1) - 1 = a(n)*(a(n)^2 - a(n) + 1) - 1 = a(n)^3 - a(n)^2 + a(n) - 1 = (a(n)^2 + 1)*(a(n) - 1) = (a(n) + a(n)^2 - a(n) + 1)*(a(n) - 1) = (a(n) + a(n+1))*(a(n) - 1). - Mohamed Bouhamida, Aug 29 2009
This sequence is also related to the short side (or hypotenuse) of adjacent right triangles, (3, 4, 5), (5, 12, 13), (13, 84, 85), ... by A053630(n) = 2*a(n) - 1. - Yuksel Yildirim, Jan 01 2013, edited by M. F. Hasler, May 19 2017
For n >= 4, a(n) mod 3000 alternates between 1807 and 2443. - Robert Israel, Jan 18 2015
The set of prime factors of a(n)'s is thin in the set of primes. Indeed, Odoni showed that the number of primes below x dividing some a(n) is O(x/(log x log log log x)). - Tomohiro Yamada, Jun 25 2018
Sylvester numbers when reduced modulo 864 form the 24-term arithmetic progression 7, 43, 79, 115, 151, 187, 223, 259, 295, 331, ..., 763, 799, 835 which repeats itself until infinity. This was first noticed in March 2018 and follows from the work of Sondow and MacMillan (2017) regarding primary pseudoperfect numbers which similarly form an arithmetic progression when reduced modulo 288. Giuga numbers also form a sequence resembling an arithmetic progression when reduced modulo 288. - Mehran Derakhshandeh, Apr 26 2019
Named after the English mathematician James Joseph Sylvester (1814-1897). - Amiram Eldar, Mar 09 2024
Guy askes if it is an irrationality sequence (see Guy, 1981). - Stefano Spezia, Oct 13 2024

			a(0)=2, a(1) = 2+1 = 3, a(2) = 2*3 + 1 = 7, a(3) = 2*3*7 + 1 = 43.

Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
Richard K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E24.
Richard K. Guy and Richard Nowakowski, Discovering primes with Euclid. Delta, Vol. 5 (1975), pp. 49-63.
Amarnath Murthy, Smarandache Reciprocal partition of unity sets and sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.1.
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..12
David Adjiashvili, Sandro Bosio and Robert Weismantel, Dynamic Combinatorial Optimization: a complexity and approximability study, 2012.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
Gennady Bachman and Troy Kessler, On divisibility properties of certain multinomial coefficients—II, Journal of Number Theory, Volume 106, Issue 1, May 2004, Pages 1-12.
Andreas Bäuerle, Sharp volume and multiplicity bounds for Fano simplices, arXiv:2308.12719 [math.CO], 2023.
Kevin S. Brown, Odd, Greedy and Stubborn (Unit Fractions).
Eunice Y. S. Chan and Robert M. Corless, Minimal Height Companion Matrices for Euclid Polynomials, Mathematics in Computer Science, Vol. 13, No. 1-2 (2019), pp. 41-56, arXiv preprint, arXiv:1712.04405 [math.NA], 2017.
Hung Viet Chu, A Threshold for the Best Two-term Underapproximation by the Greedy Algorithm, arXiv:2306.12564 [math.NT], 2023.
Matthew Brendan Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly, Vol. 29, No. 10 (1922), pp. 380-387.
Mehran Derakhshandeh, Why do Sylvester numbers, when reduced modulo 864, form an arithmetic progression 7,43,79,115,151,187,223,...?
Paul Erdős and E. G. Straus, On the Irrationality of Certain Ahmes Series, J. Indian Math. Soc. (N.S.), 27(1964), pp. 129-133.
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.
Solomon W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math., 15 (1963), 475-478.
Solomon W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
Richard K. Guy and Richard Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
János Kollár, Which powers of holomorphic functions are integrable?, arXiv:0805.0756 [math.AG], 2008.
E. Lemoine, Sur la décomposition d'un nombre en ses carrés maxima, Assoc. Française pour L'Avancement des Sciences (1896), 73-77.
Zheng Li and Quanyu Tang, On a conjecture of Erdős and Graham about the Sylvester's sequence, arXiv:2503.12277 [math.NT], 2025. See p. 2.
Nick Lord, A uniform construction of some infinite coprime sequences, The Mathematical Gazette, vol. 92, no. 523, March 2008, pp.66-70.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Melvyn B. Nathanson, Underapproximation by Egyptian fractions, arXiv:2202.00191 [math.NT], 2022.
Benjamin Nill, Volume and lattice points of reflexive simplices, Discrete & Computational Geometry, Vol. 37, No. 2 (2007), pp. 301-320, arXiv preprint, arXiv:math/0412480 [math.AG], 2004-2007.
R. W. K. Odoni, On the prime divisors of the sequence w_{n+1}=1+w_1 ... w_n, J. London Math. Soc. 32 (1985), 1-11.
Michael Penn, An intriguing integer sequence — Sylvester’s Sequence, YouTube video (2022).
Simon Plouffe, A set of formulas for primes, arXiv:1901.01849 [math.NT], 2019.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 5. [?Broken link]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 5.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Vol. 113, No. 10 (Dec., 2006), pp. 937-938.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240, arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
J. J. Sylvester, On a Point in the Theory of Vulgar Fractions, Amer. J. Math., Vol. 3, No. 4 (1880), pp. 332-335.
Burt Totaro, The ACC conjecture for log canonical thresholds, Séminaire Bourbaki no. 1025 (juin 2010).
Burt Totaro and Chengxi Wang, Varieties of general type with small volume, arXiv:2104.12200 [math.AG], 2021.
Akiyoshi Tsuchiya, The delta-vectors of reflexive polytopes and of the dual polytopes, Discrete Mathematics, Vol. 339, No. 10 (2016), pp. 2450-2456, arXiv preprint, arXiv:1411.2122 [math.CO], 2014-2016.
Stephan Wagner and Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Quadratic Recurrence Equation.
Eric Weisstein's World of Mathematics, Sylvester's Sequence.
Wikipedia, Sylvester's sequence.
Bowen Yao, A note on fraction decompositions of integers, The American Mathematical Monthly, 127(10), 928-932, Dec 2020.
Index entries for sequences of form a(n+1)=a(n)^2 + ....
Index entries for "core" sequences.

Cf. A005267, A000945, A000946, A005265, A005266, A075442, A007018, A014117, A054377, A002061, A118227, A126263, A007996 (primes dividing some term), A323605 (smallest prime divisors), A091335 (number of prime divisors), A129871 (a variant starting with 1).

a000058 0 = 2
a000058 n = a000058 m ^ 2 - a000058 m + 1 where m = n - 1
-- James Spahlinger, Oct 09 2012

a000058_list = iterate a002061 2  -- Reinhard Zumkeller, Dec 18 2013

a(n) = n == 0 ? BigInt(2) : a(n - 1)*(a(n - 1) - 1) + 1
[a(n) for n in 0:8] |> println # Peter Luschny, Dec 15 2020

A[0]:= 2:
for n from 1 to 12 do
A[n]:= A[n-1]^2 - A[n-1]+1
od:
seq(A[i],i=0..12); # Robert Israel, Jan 18 2015

a[0] = 2; a[n_] := a[n - 1]^2 - a[n - 1] + 1; Table[ a[ n ], {n, 0, 9} ]
NestList[#^2-#+1&,2,10] (* Harvey P. Dale, May 05 2013 *)
RecurrenceTable[{a[n + 1] == a[n]^2 - a[n] + 1, a[0] == 2}, a, {n, 0, 10}] (* Emanuele Munarini, Mar 30 2017 *)

a(n) := if n = 0 then 2 else a(n-1)^2-a(n-1)+1 $
makelist(a(n),n,0,8); /* Emanuele Munarini, Mar 23 2017 */

a(n)=if(n<1,2*(n>=0),1+a(n-1)*(a(n-1)-1))

A000058(n,p=2)={for(k=1,n,p=(p-1)*p+1);p} \\ give Mod(2,m) as 2nd arg to calculate a(n) mod m. - M. F. Hasler, Apr 25 2014

a=vector(20); a[1]=3; for(n=2, #a, a[n]=a[n-1]^2-a[n-1]+1); concat(2, a) \\ Altug Alkan, Apr 04 2018

A000058 = [2]
for n in range(1, 10):
    A000058.append(A000058[n-1]*(A000058[n-1]-1)+1)
# Chai Wah Wu, Aug 20 2014

a(n) = 1 + a(0)*a(1)*...*a(n-1).
a(n) = a(n-1)*(a(n-1)-1) + 1; Sum_{i>=0} 1/a(i) = 1. - Néstor Romeral Andrés, Oct 29 2001
Vardi showed that a(n) = floor(c^(2^(n+1)) + 1/2) where c = A076393 = 1.2640847353053011130795995... - Benoit Cloitre, Nov 06 2002 (But see the Aho-Sloane paper!)
a(n) = A007018(n+1)+1 = A007018(n+1)/A007018(n) [A007018 is a(n) = a(n-1)^2 + a(n-1), a(0)=1]. - Gerald McGarvey, Oct 11 2004
a(n) = sqrt(A174864(n+1)/A174864(n)). - Giovanni Teofilatto, Apr 02 2010
a(n) = A014117(n+1)+1 for n = 0,1,2,3,4; a(n) = A054377(n)+1 for n = 1,2,3,4. - Jonathan Sondow, Dec 07 2013
a(n) = f(1/(1-(1/a(0) + 1/a(1) + ... + 1/a(n-1)))) where f(x) is the smallest integer > x (see greedy algorithm above). - Robert FERREOL, Feb 22 2019
From Amiram Eldar, Oct 29 2020: (Start)
Sum_{n>=0} (-1)^n/(a(n)-1) = A118227.
Sum_{n>=0} (-1)^n/a(n) = 2 * A118227 - 1. (End)

A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

Original entry on oeis.org

1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 0

N. J. A. Sloane

Number of ordered trees having nodes of outdegree 0,1,2 and such that all leaves are at level n. Example: a(2)=6 because, denoting by I a path of length 2 and by Y a Y-shaped tree with 3 edges, we have I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots. - Emeric Deutsch, Oct 31 2002
Equivalently, the number of acyclic digraphs (dags) that unravel to a perfect binary tree of height n. - Nachum Dershowitz, Jul 03 2022
a(n) has at least n different prime factors. [Saidak]
Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 [This has been questioned, see MathOverflow link. - Charles R Greathouse IV, Mar 30 2015]
For prime factors see A007996.
Curtiss shows that if the reciprocal sum of the multiset S = {x_1, x_2, ..., x_n} is 1, then max(S) <= a(n). - Charles R Greathouse IV, Feb 28 2007
The number of reduced ZBDDs for Boolean functions of n variables in which there is no zero sink. (ZBDDs are "zero-suppressed binary decision diagrams.") For example, a(2)=6 because of the 2-variable functions whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111. - Don Knuth, Jun 04 2007
Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that a(n) is the integer just a tiny bit below the real number theta^{2^n}-1/2, where theta =~ 1.597910218 is the exponential of the rapidly convergent series Sum_{n>=0} log(1+1/a_n)/2^{n+1}. For example, theta^32 - 1/2 =~ 3263442.0000000383. - Don Knuth, Jun 04 2007 [Corrected by Darryl K. Nester, Jun 19 2017]
The next term has 209 digits. - Harvey P. Dale, Sep 07 2011
Urquhart shows that a(n) is the minimum size of a tableau refutation of the clauses of the complete binary tree of depth n, see pp. 432-434. - Charles R Greathouse IV, Jan 04 2013
For any positive a(0), the sequence a(n) = a(n-1) * (a(n-1) + 1) gives a constructive proof that there exists integers with at least n distinct prime factors, e.g. a(n). As a corollary, this gives a constructive proof of Euclid's theorem stating that there are an infinity of primes. - Daniel Forgues, Mar 03 2017
Lower bound for A100016 (with equality for the first 5 terms), where a(n)+1 is replaced by nextprime(a(n)). - M. F. Hasler, May 20 2019

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
David Adjiashvili, Sandro Bosio and Robert Weismantel, Dynamic Combinatorial Optimization: a complexity and approximability study, 2012.
Gilles Audemard, Steve Bellart, Louenas Bounia, Frédéric Koriche, Jean-Marie Lagniez, and Pierre Marquis, On the Explanatory Power of Decision Trees, arXiv:2108.05266 [cs.AI], 2021.
Arvind Ayyer, Anne Schilling, Benjamin Steinberg and Nicolas M. Thiéry, Markov chains, R-trivial monoids and representation theory, Int. J. Algebra Comput., Vol. 25 (2015), pp. 169-231, arXiv preprint, arXiv:1401.4250 [math.CO], 2014.
Umberto Cerruti, Percorsi tra i numeri (in Italian), page 5.
A. Yu. Chirkov, D. V. Gribanov and N. Yu. Zolotykh, On the Proximity of the Optimal Values of the Multi-Dimensional Knapsack Problem with and without the Cardinality Constraint, arXiv:2004.08589 [math.OC], 2020.
D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly, Vol. 29, No. 10 (1922), pp. 380-387.
Christian Elsholtz and Stefan Planitzer, Sums of four and more unit fractions and approximate parametrizations, arXiv:2012.05984 [math.NT], 2020.
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 9.
Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.
Diana Maimuţ and George Teşeleanu, Inferring Bivariate Polynomials for Homomorphic Encryption Application, Cryptology ePrint Archive (2023) Art. 844. See p. 16.
MathOverflow, Is OEIS A007018 really a subsequence of squarefree numbers?.
Marko R. Riedel, Two-colorings of unordered full binary trees on n levels.
Matthew Roughan, Surreal Birthdays and Their Arithmetic, arXiv:1810.10373 [math.HO], 2018.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Vol. 113, No. 10 (Dec., 2006), pp. 937-938.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
Alasdair Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, Vol. 1, No. 4 (1995) pp. 425-467, esp. p. 434.
Zalman Usiskin, Letter to N. J. A. Sloane, Oct. 1991.
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Cf. A000058, A003687, A004168, A011782, A077125, A117805, A118227, A371321.
Lower bound for A100016.
Row sums of A122888.

a007018 n = a007018_list !! n
a007018_list = iterate a002378 1  -- Reinhard Zumkeller, Dec 18 2013

[n eq 1 select 1 else Self(n-1)^2 + Self(n-1): n in [1..10]]; // Vincenzo Librandi, May 19 2015

A007018 := proc(n)
    option remember;
    local aprev;
    if n = 0 then
        1;
    else
        aprev := procname(n-1) ;
        aprev*(aprev+1) ;
    end if;
end proc: # R. J. Mathar, May 06 2016

FoldList[#^2 + #1 &, 1, Range@ 8] (* Robert G. Wilson v, Jun 16 2011 *)
NestList[#^2 + #&, 1, 10] (* Harvey P. Dale, Sep 07 2011 *)

a[1]:1$
a[n]:=(a[n-1] + (a[n-1]^2))$
A007018(n):=a[n]$
makelist(A007018(n),n,1,10); /* Martin Ettl, Nov 08 2012 */

a(n)=if(n>0,my(x=a(n-1));x^2+x,1) \\ Edited by M. F. Hasler, May 20 2019 and Jason Yuen, Mar 01 2025

from itertools import islice
def A007018_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a *= a+1
A007018_list = list(islice(A007018_gen(),9)) # Chai Wah Wu, Mar 19 2024

a(n) = A000058(n)-1 = A000058(n-1)^2 - A000058(n-1) = 1/(1-Sum_{jA000058(j)) where A000058 is Sylvester's sequence. - Henry Bottomley, Jul 23 2001
a(n) = floor(c^(2^n)) where c = A077125 = 1.597910218031873178338070118157... - Benoit Cloitre, Nov 06 2002
a(1)=1, a(n) = Product_{k=1..n-1} (a(k)+1). - Benoit Cloitre, Sep 13 2003
a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller, Apr 10 2008
If an (additional) initial 1 is inserted, a(n) = Sum_{kFranklin T. Adams-Watters, Jun 11 2009
a(n+1) = a(n)-th oblong (or promic, pronic, or heteromecic) numbers (A002378). a(n+1) = A002378(a(n)) = A002378(a(n-1)) * (A002378(a(n-1)) + 1). - Jaroslav Krizek, Sep 13 2009
a(n) = A053631(n)/2. - Martin Ettl, Nov 08 2012
Sum_{n>=0} (-1)^n/a(n) = A118227. - Amiram Eldar, Oct 29 2020
Sum_{n>=0} 1/a(n) = A371321. - Amiram Eldar, Mar 19 2024

A126263 List of primes generated by factoring successive integers in Sylvester's sequence (A000058).

Original entry on oeis.org

2, 3, 7, 43, 13, 139, 3263443, 547, 607, 1033, 31051, 29881, 67003, 9119521, 6212157481, 5295435634831, 31401519357481261, 77366930214021991992277, 181, 1987, 112374829138729, 114152531605972711, 35874380272246624152764569191134894955972560447869169859142453622851

Offset: 1

Howard L. Warth (hlw6c2(AT)umr.edu), Dec 22 2006

nonn

The list is infinite and no term repeats since Sylvester's sequence is an infinite coprime sequence.

However, it appears to be unknown whether all terms in A000058 are squarefree. - Jeppe Stig Nielsen, Apr 23 2020

			2 = 2, 3 = 3, 7 = 7, 43 = 43, 1807 = 13 * 139, 3263443 = 3263443,
10650056950807 = 547 * 607 * 1033 * 31051,
113423713055421844361000443 = 29881 * 67003 * 9119521 * 6212157481,
12864938683278671740537145998360961546653259485195807 = 5295435634831 * 31401519357481261 * 77366930214021991992277.
165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 = 181 * 1987 * 112374829138729 * 114152531605972711 * 35874380272246624152764569191134894955972560447869169859142453622851. - _Jonathan Sondow_, Jan 26 2014

Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 9.

Ray Chandler, Table of n, a(n) for n = 1..28 (first 27 terms from William Stein)
J. K. Andersen, Factorization of Sylvester's sequence.
Filip Saidak, Proof of Euclid's Theorem.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec. 2006.

Cf. A000058, A007996, A236433.

a(0):=2; for n from 0 to 8 do a(n+1):=a(n)^2-a(n)+1;ifactor(%); od;

Flatten[FactorInteger[NestList[#^2 - # + 1 &, 2, 8]][[All, All, 1]]] (* Paolo Xausa, Sep 09 2024 *)

v=[2]; for(i=1,10, v=concat(v,Set(factor(vecprod(v)+1)[,1]))); v \\ Charles R Greathouse IV, Oct 02 2014

v = [2]
for n in range(12):
    v.append(v[-1]^2-v[-1]+1)
    print(prime_divisors(v[-1])) # William Stein, Aug 26 2009

Offset corrected by N. J. A. Sloane, Aug 20 2009
a(23)-a(27) from William Stein (wstein(AT)gmail.com), Aug 20 2009, Aug 21 2009
a(17) corrected by D. S. McNeil, Dec 10 2010
b-file updated at the suggestion of Hans Havermann by Ray Chandler, Feb 27 2015

A362250 Primes dividing terms of A231831.

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 89, 101, 137, 157, 211, 373, 659, 877, 881, 1399, 1597, 1627, 1663, 1811, 2029, 2069, 2087, 2153, 2381, 2677, 2939, 3433, 3491, 3511, 3617, 3673, 4111, 4127, 4547, 4721, 5059, 5483, 6529, 6793, 6827, 7757, 8209, 8297, 8677, 9203, 9463, 9811, 10139, 10159, 11321

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Since the terms of A231831 are pairwise coprime, each prime divides at most one term of A231831. Indices of the corresponding terms are listed in A362251, and so a(n) divides A231831(A362251(n)).

Max Alekseyev, Table of n, a(n) for n = 1..16944
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A007996, A180871, A231830, A231831, A362251, A362252, A362253.

A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 9, 4, 4, 8, 3, 31, 12, 7, 7, 9, 44, 8, 22, 29, 36, 37, 8, 21, 5, 26, 4, 20, 24, 12, 76, 30, 5, 47, 5, 13, 9, 25, 6, 41, 51, 9, 53, 6, 27, 39, 5, 12, 78, 64, 10, 185, 113, 205, 91, 85, 43, 195, 32, 117, 20, 133, 142, 119, 64, 70, 199, 41, 125, 79, 243, 70, 35, 105, 67, 156

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..16944

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362252, A362253.

A362252 Primes dividing terms of A231830.

Original entry on oeis.org

5, 53, 89, 101, 373, 877, 1109, 1181, 1597, 1613, 2029, 2069, 2153, 2213, 2381, 2741, 3617, 4273, 6529, 6737, 7417, 7717, 11321, 12653, 13009, 13309, 16829, 17729, 23581, 23993, 25373, 32569, 33353, 33857, 34841, 35053, 36097, 37201, 38609, 41513, 42461, 48661, 55829, 58369, 59093, 63281

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Since the terms of A231830 are pairwise coprime, each prime divides at most one term of A231830. Indices of the corresponding terms are listed in A362253, and so a(n) divides A231830(A362253(n)).

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362253.

A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

Original entry on oeis.org

1, 4, 7, 2, 19, 25, 30, 38, 45, 4, 26, 33, 27, 46, 10, 59, 102, 38, 84, 37, 22, 77, 80, 37, 240, 57, 45, 240, 173, 38, 41, 100, 88, 44, 114, 39, 63, 24, 14, 121, 177, 12, 155, 270, 65, 109, 44, 391, 54, 22, 96, 320, 194, 347, 182, 226, 143, 290, 105, 135, 29, 198, 113, 302, 572, 53, 692, 168, 366

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362252.

A096264 Primes that do not divide any terms of the sequence f given by f(1) = 2, f(n+1) = f(n)^2-f(n)+1.

Original entry on oeis.org

5, 11, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Jeffrey Shallit, Aug 04 2004

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..5000
R. W. K. Odoni, On the prime divisors of the sequence w_{n+1} = 1+w_1 ... w_n, J. London Math. Soc. 32 (1985), 1-11.

Complement of A007996 (in the primes).

IsSylvester:=proc(p)
local x,S ;x:=2 : S:=NULL: while not member(x, [S]) do if x=0 then
return(true) fi; S := S ,x; x := (x^2-x+1) mod p od : return false end:
L:=NULL:p:=5:while p<1000 do if not IsSylvester(p) then L:=L,p fi : p:=nextprime(p) od:
L;
# Robert FERREOL, Feb 14 2019

Definition corrected (following a remark by Don Reble) by M. F. Hasler, Apr 24 2014

cp's OEIS Frontend

A180871 Index of term in Sylvester's sequence A000058 divisible by prime A007996(n).

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A096263 Duplicate of A007996.

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A000058 Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.

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A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

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A126263 List of primes generated by factoring successive integers in Sylvester's sequence (A000058).

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A362250 Primes dividing terms of A231831.

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A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

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