A126263 -id:A126263 - cp's OEIS frontend

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

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.
Thomas Bloom, Problem 243 and Problem 315, Erdős Problems.
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.
Terence Tao, Erdős problem database, see nos. 243, 315.
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)

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

Original entry on oeis.org

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

A236434 Table whose n-th row lists the prime factors of the n-th Giuga number A007850(n).

Original entry on oeis.org

2, 3, 5, 2, 3, 11, 13, 2, 3, 7, 41, 2, 3, 11, 17, 59, 2, 3, 11, 23, 31, 47057, 2, 3, 7, 43, 3041, 4447, 2, 3, 7, 59, 163, 1381, 775807, 2, 3, 7, 71, 103, 67213, 713863, 2, 3, 7, 71, 103, 61559, 29133437, 2, 3, 11, 23, 31, 47137, 28282147, 3892535183, 2, 3, 11, 23, 31, 47059, 2259696349, 110725121051, 2, 3, 7, 43, 1831, 138683, 2861051, 1456230512169437

Offset: 1

Jonathan Sondow, Jan 25 2014

It is unknown whether there are infinitely many Giuga numbers or any odd ones.
See A007850 for other comments, references, links, etc.
For prime factors of primary pseudoperfect numbers, see  A236433; of terms in Sylvester's sequence, see A126263.

			30 = 2 * 3 * 5.
858 = 2 * 3 * 11 * 13.
1722 = 2 * 3 * 7 * 41.
66198 = 2 * 3 * 11 * 17 * 59.
2214408306 = 2 * 3 * 11 * 23 * 31 * 47057.
24423128562 = 2 * 3 * 7 * 43 * 3041 * 4447.
432749205173838 = 2 * 3 * 7 * 59 * 163 * 1381 * 775807.
14737133470010574 = 2 * 3 * 7 * 71 * 103 * 67213 * 713863.
550843391309130318 = 2 * 3 * 7 * 71 * 103 * 61559 * 29133437.
244197000982499715087866346 = 2 * 3 * 11 * 23 * 31 * 47137 * 28282147 * 3892535183.
554079914617070801288578559178 = 2 * 3 * 11 * 23 * 31 * 47059 * 2259696349 * 110725121051.
1910667181420507984555759916338506 = 2 * 3 * 7 * 43 * 1831 * 138683 * 2861051 * 1456230512169437.
Another Giuga number (but possibly not the 13th) is 4200017949707747062038711509670656632404195753751630609228764416142557211582098432545190323474818 = 2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519 * 8491659218261819498490029296021 * 58254480569119734123541298976556403.

M. F. Hasler, Table of n, a(n) for n = 1..73

Cf. A007850, A216823, A216824, A236433.

A236433 List of primes generated by factoring successive primary pseudoperfect numbers (A054377).

Original entry on oeis.org

2, 2, 3, 2, 3, 7, 2, 3, 7, 43, 2, 3, 11, 23, 31, 2, 3, 11, 23, 31, 47059, 2, 3, 11, 17, 101, 149, 3109, 2, 3, 11, 23, 31, 47059, 2217342227, 1729101023519

Offset: 1

Jonathan Sondow, Jan 25 2014

It is unknown whether there are infinitely many primary pseudoperfect numbers or any odd ones.
See A054377 for other comments, references, links, etc.
For prime factors of Giuga numbers, see A236434; of terms in Sylvester's sequence, see A126263.

			2 = 2.
6 = 2 * 3.
42 = 2 * 3 * 7.
1806 = 2 * 3 * 7 * 43.
47058 = 2 *  3 * 11 * 23 * 31.
2214502422 =  2 *  3 * 11 * 23 * 31 * 47059.
52495396602 =  2 * 3 * 11 * 17 * 101 * 149 * 3109.
8490421583559688410706771261086 = 2 *  3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519.

Cf. A054377, A126263, A216825, A216826, A236434.

A054377 = Cases[Import["https://oeis.org/A054377/b054377.txt", "Table"], {, }][[All, 2]];
First /@ Flatten[FactorInteger[A054377], 1] (* Robert Price, Mar 14 2020 *)

A375543 Sylvester primes. Yet another proof of the infinity of primes.

Original entry on oeis.org

2, 3, 7, 43, 13, 139, 547, 607, 1033, 181, 1987, 73, 2287, 29881, 13999, 17881, 31051, 52387, 67003, 74203, 128551, 352867, 635263, 74587, 1286773, 2271427, 27061, 164299, 20929, 1171, 298483, 1679143, 3229081, 3263443, 120823, 447841, 2408563, 333457, 30241, 4219

Offset: 1

Peter Luschny, Sep 02 2024

nonn

Sylvester's sequence can be defined recursively S(n) = S(n-1)*(S(n-1) + 1) for n >= 1 starting S(0) = 1. (A000058(n) = S(n) + 1.)
Since S(n) and S(n) + 1 have no common divisors, it follows that S(n) has at least one more prime factor than S(n-1), and thus by induction, S(n) has at least n distinct prime factors. This simple and constructive form of Euclid's proof of the infinity of primes was formulated by Filip Saidak (see links).
To generate the sequence, select the smallest unchosen prime factor from all prime factors of S(0), S(1), ..., S(n-1). We call the infinite sequence constructed this way the 'Sylvester primes'. The terms, when ordered by size, can be found in A007996; any prime not a Sylvester prime can be located in A096264.
As a procedure, the sequence can hardly be described more clearly than in the Maple program below. Compared to other variants (for example A126263), it has the advantage that the primes generated start relatively small.

			The generation of the sequence starts:
  n   selected       factors of S(i), i<n       a(n)
  [1] {}             {2}                     ->   2,
  [2] {2}            {2, 3}                  ->   3,
  [3] {2, 3}         {2, 3, 7}               ->   7,
  [4] {2, 3, 7}      {2, 3, 7, 43}           ->  43,
  [5] {2, 3, 7, 43}  {2, 3, 7, 13, 43, 139}  ->  13.

Jens Kruse Andersen, Factorization of Sylvester's sequence.
Chris Caldwell, Filip Saidak's Proof, PrimePages.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Vol. 113, No. 10 (Dec., 2006), pp. 937-938.

Cf. A000040, A000058, A007018, A007996, A014546, A091336, A096264, A375545 (indices in P).

Variants: A126263, A367020.

fact := n -> NumberTheory:-PrimeFactors(n):
SylvesterPrimes := proc(len) local p, d, w, n;
p := 1; d := {}; w := {};
for n from 1 to len do
   p := p*(p + 1);
   d := fact(p) minus w;
   w := w union {min(d)};
od end:
SylvesterPrimes(8);
isSylvesterPrime := proc(p) local s, M;
M := NULL: s := 2:
while not member(s, [M]) do
   M := M, s;
   s := (s^2 + s) mod p;
   if s = 0 then return true fi;
od: false end:

Module[{nmax = 20, a = {}, p = 1, f}, Do[p *= p + 1; f = 2; While[MemberQ[a,f] || !Divisible[p, f], f = NextPrime[f]]; AppendTo[a, f], nmax]; a] (* Paolo Xausa, Sep 03 2024 *)

from sympy import sieve
from itertools import count, islice
def smallest_new_primefactor(n, pf):
    return next(pi for i in count(1) if (pi:=sieve[i]) not in pf and n%pi==0)
def agen(): # generator of terms
    p, d, w, pf = 1, set(), set(), set()
    while True:
        p = p*(p + 1)
        m = smallest_new_primefactor(p, w)
        w |= {m}
        yield m
print(list(islice(agen(), 20)))
# Michael S. Branicky, Sep 02 2024 after Peter Luschny

# Returns the first 24 terms in less than 60 seconds.
def SylvesterPrimes(len: int) -> list[int]:
    M: list[int] = []
    p = q = 1
    for n in range(1, len + 1):
        p = p * (p + 1)
        pq = p // q
        for s in Primes():
            if pq % s == 0 and not s in M:
                M.append(s)
                q = q * s
                print(n, s)
                break
    return M
SylvesterPrimes(24)  # Peter Luschny, Sep 05 2024

a(21)-a(31) from Michael S. Branicky, Sep 03 2024
a(32) from Paolo Xausa, Sep 04 2024
a(33)-a(36) from Peter Luschny, Sep 05 2024
a(37)-a(40) from Jinyuan Wang, Jul 25 2025

A256147 First repeated number in Sylvester's sequence modulo prime(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 7, 3, 2, 6, 2, 1, 7, 7, 7, 17, 7, 3, 1, 43, 66, 2, 72, 51, 7, 50, 32, 3, 111, 85, 26, 1, 44, 31, 7, 7, 96, 157, 23, 1, 88, 3, 97, 7

Offset: 1

Alonso del Arte, Mar 16 2015

nonn

Sylvester's sequence (A000058) is an infinite coprime sequence, a fact that may lead to the incorrect intuition that all primes occur as factors of its terms. It's quite easy to check that no multiple of 5 occurs, since Sylvester's sequence modulo 5 is 2, 3, 2, 3, 2, 3, ...
If a multiple of p occurs in Sylvester's sequence at position j, then A000058(k) == 1 (mod p) for all k > j.
But if no multiple of p occurs in Sylvester's sequence at all, then Sylvester's sequence is fully periodic modulo p or it enters a cycle at some point.

			a(4) = 1, because the fourth prime is 7 and Sylvester's sequence modulo 7 is 2, 3, 0, 1, 1, 1, ...
a(5) = 3, because the fifth prime is 11 and Sylvester's sequence modulo 11 is 2, 3, 7, 10, 3, 7, 10, 3, 7, 10, ... (3 is the first number repeated).

J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 388 - 389.

Cf. A007996, A126263, A255595.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Haskell

Julia

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A236434 Table whose n-th row lists the prime factors of the n-th Giuga number A007850(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A236433 List of primes generated by factoring successive primary pseudoperfect numbers (A054377).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A375543 Sylvester primes. Yet another proof of the infinity of primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Extensions

A256147 First repeated number in Sylvester's sequence modulo prime(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs