0,1

Also called Euclid numbers.

Another version begins 1, 2, 3, 7, 43, 1807, ..., but the initial 1 seems artificial.

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)x1 rectangle which can be divided in 3 squares. Now you have a 6x1 rectangle at the bottom. Instead of dividing it into 6 squares, draw another horizontal line so you obtain a (6+1=7)x1 rectangle and a 42x1 rectangle left. Etc... - Nestor Romeral Andres (cashogor(AT)yahoo.com), 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(1)+1/a(2)+...1/a(n-1) = (a(n)-2)/(a(n)-1). Thus we can also write a(n) = ( 1/a(1)+1/a(2)+...1/a(n-1)-2 )/( 1/a(1)+1/a(2)+...1/a(n-1)-1 ). - Frederick Magata (fmagata(AT)mi.uni-koeln.de), May 10 2001

A greedy sequence: a(n+1) is the smallest integer > a(n) such that 1/a(1) + 1/a(2) + ... + 1/a(n+1) doesn't exceed 1. - Ulrich Schimke, Nov 17, 2002

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

Consider the mapping f(a/b) = (a^3 + b)/(a +b^3). Taking a = 1 b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/2,1/3,4/28=1/7,8/344=1/43,... 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 (amarnath_murthy(AT)yahoo.com), 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 (amarnath_murthy(AT)yahoo.com), 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 (davidwwilson(AT)comcast.net), 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 -ma(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).

D. R. Curtiss, "On Kellogg's Diophantine problem" Amer Math. Monthly, 29, (1922), 380-387.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.

S. W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math., 15 (1963), 475-478.

S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.

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.

Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006

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, Fib. Quart., 11 (1973), 429-437.

K. S. Brown, Odd, Greedy, and Stubborn (Unit Fractions)

B. Nill, Volume and lattice points of reflexive simplices

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 5.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Quadratic Recurrence Equation

Wikipedia, Sylvester's sequence

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Index entries for "core" sequences

a(n) = 1 + a(0)*a(1)*...*a(n-1).

a(n) = a(n-1)*(a(n-1)-1)+1; Sum(i=0 to oo) 1/a(i) = 1. - Nestor Romeral Andres (cashogor(AT)yahoo.com), Oct 29 2001

Vardi showed that a(n) = floor(c^(2^(n+1))+1/2) where c=1.2640847353053011130795995... - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (Gerald.McGarvey(AT)comcast.net), Oct 11 2004

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

a[0] = 2; a[n_] := a[n - 1]^2 - a[n - 1] + 1; Table[ a[ n ], {n, 0, 9} ]

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

Cf. A005267, A000945, A000946, A005265, A005266, A075442, A007018.

Adjacent sequences: A000055 A000056 A000057 this_sequence A000059 A000060 A000061

Sequence in context: A113845 A072713 A129871 this_sequence A075442 A082993 A071580

nonn,nice,core

njas