1,1

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.

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.

R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag, NY, Berlin, 1994, A10.

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

Chris Caldwell, The Prime Glossary, Goldbach's conjecture.

Andrew M. Odlyzko, Iterated Absolute Values of Differences of Consecutive Primes, Math. Comp. 61 (1993), 373-380.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

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

See the triangle in A036262.

a = {}; l = Table[ Prime[n], {n, 111}]; Do[l = Abs[ Drop[l, 1] - Drop[l, -1]]; AppendTo[ a, l[[2]]], {n, 1, 109}]; a

See A036262 for an abbreviated table of absolute differences.

Sequence in context: A023556 A044944 A044945 this_sequence A044946 A044947 A044948

Adjacent sequences: A089579 A089580 A089581 this_sequence A089583 A089584 A089585

easy,nonn

Robert G. Wilson v (rgwv(AT)rgwv.com) and R. K. Guy (rkg(AT)cpsc.ucalgary.ca), Nov 08 2003