TOPICS
Search

Sylvester's Sequence


The sequence defined by e_0=2 and the quadratic recurrence equation

 e_n=1+product_(i=0)^(n-1)e_i=e_(n-1)^2-e_(n-1)+1.
(1)

This sequence arises in Euclid's proof that there are an infinite number of primes. The proof proceeds by constructing a sequence of primes using the recurrence relation

 e_(n+1)=e_0e_1...e_n+1
(2)

(Vardi 1991). Amazingly, there is a constant

 E=1/2sqrt(6)exp{sum_(j=1)^infty2^(-j-1)ln[1+(2e_j-1)^(-2)]}=1.2640847353...
(3)

(OEIS A076393) such that

 e_n=|_E^(2^(n+1))+1/2_|
(4)

(Aho and Sloane 1973, Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (OEIS A000058). The e_n satisfy

 sum_(n=0)^infty1/(e_n)=1.
(5)

In addition, if 0<x<1 is an irrational number, then the nth term of an infinite sum of unit fractions used to represent x as computed using the greedy algorithm must be smaller than 1/e_n.

The n of the first few prime e_n are 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443, ... (OEIS A014546). Vardi (1991) gives a lists of factors less than 5×10^7 of e_n for n<=200 and shows that e_n is composite for 6<=n<=17. Furthermore, all numbers less than 2.5×10^(15) in Sylvester's sequence are squarefree, and no squareful numbers in this sequence are known (Vardi 1991).


See also

Cahen's Constant, Euclid's Theorems, Greedy Algorithm, Quadratic Recurrence Equation, Squarefree, Squareful

Explore with Wolfram|Alpha

References

Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart. 11, 429-437, 1973.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Research problem 4.65 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Sloane, N. J. A. Sequences A000058/M0865, A014546, and A076393 in "The On-Line Encyclopedia of Integer Sequences."Finch, S. R. "Quadratic Recurrence Constants." §6.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 443-448, 2003.Vardi, I. "Are All Euclid Numbers Squarefree?" and "PowerMod to the Rescue." §5.1 and 5.2 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89, 1991.

Referenced on Wolfram|Alpha

Sylvester's Sequence

Cite this as:

Weisstein, Eric W. "Sylvester's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SylvestersSequence.html

Subject classifications