%I A089582
%S A089582 2,0,2,2,2,2,2,2,0,0,0,0,0,0,2,2,0,2,2,0,0,2,2,2,0,0,0,2,2,0,2,0,0,0,2,
%T A089582 2,0,0,0,0,0,0,2,2,0,2,2,0,2,0,0,2,0,2,2,2,2,0,0,0,0,0,0,2,0,0,2,2,0,0,
%U A089582 2,2,0,2,0,0,0,0,0,2,0,2,2,2,2,2,0,0,2,2,0,0,2,2,0,0,0,0,2,2,2,2,0,0,0
%N A089582 From Gilbreath's conjecture.
%C A089582 Let d_0(n) = p_n, the n-th prime, for n = 1 and let d_k+1 (n) = | d_k(n) 
               - d_k(n+1) | for k = 0, n = 1. A well known conjecture, usually ascribed 
               to Gilbreath but actually due to Proth in the 19-th century, says 
               that d_k(1) = 1 for all k >= 1. This sequence gives d_k(2) for all 
               k >1 and for the conjecture to be true, this sequence must contain 
               only 0's and 2's. Although not necessary to the conjecture's validity, 
               the 0's and 2's are of roughly equal count.
%C A089582 The paper cited below by A. M. Odlyzko reports on a computation that 
               verified this conjecture for k = p(1013) ~ 3 * 10^11. It also discusses 
               the evidence and the heuristics about this conjecture. It is very 
               likely that similar conjectures are also valid for many other integer 
               sequences.
%D A089582 R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag, 
               NY, Berlin, 1994, A10.
%D A089582 P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, 
               New York, NY, pp. xxiv+541, ISBN 0-387-94457-5. 1995. MR 96k:11112
%H A089582 Chris Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=GilbreathsConjecture">
               The Prime Glossary, Goldbach's conjecture</a>.
%H A089582 Andrew M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/arch/
               gilbreath.conj.ps">Iterated Absolute Values of Differences of Consecutive 
               Primes</a>, Math. Comp. 61 (1993), 373-380.
%H A089582 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A089582 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GilbreathsConjecture.html">Link to a section of The World of Mathematics.</
               a>.
%e A089582 See the triangle in A036262.
%t A089582 a = {}; l = Table[ Prime[n], {n, 111}]; Do[l = Abs[ Drop[l, 1] - Drop[l, 
               -1]]; AppendTo[ a, l[[2]]], {n, 1, 109}]; a
%Y A089582 See A036262 for an abbreviated table of absolute differences.
%Y A089582 Sequence in context: A023556 A044944 A044945 this_sequence A044946 A044947 
               A044948
%Y A089582 Adjacent sequences: A089579 A089580 A089581 this_sequence A089583 A089584 
               A089585
%K A089582 easy,nonn
%O A089582 1,1
%A A089582 Robert G. Wilson v (rgwv(AT)rgwv.com) and R. K. Guy (rkg(AT)cpsc.ucalgary.ca), 
               Nov 08 2003