Increasing integer sequences and Goldbach's conjecture

Torelli, Mauro

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 107-121.

Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.

[1] P. Capell and T.V. Narayana, On Knock-out Tournaments. Canad. Math. Bull. 13 (1970) 105-109. | Zbl

[2] L. Carlitz, D.P. Roselle and R.A. Scoville, Some Remarks on Ballot-Type Sequences of Positive Integers. J. Combinatorial Theory, Ser. A 11 (1971) 258-271. | Zbl

[3] L. Comtet, Advanced Combinatorics. Reidel, Dordrecht (1974). | MR | Zbl

[4] M. Cook and M. Kleber, Tournament sequences and Meeussen sequences. Electron. J. Combin. 7 (2000), Research Paper 44, p. 16 (electronic). | MR | Zbl

[5] J-M. Deshouillers, H.J.J. te Riele and Y. Saouter, New experimental results concerning the Goldbach conjecture, in Proc. 3rd Int. Symp. on Algorithmic Number Theory. Lect. Notes Comput. Sci. 1423 (1998) 204-215. | Zbl

[6] P. Dusart, The $k$th prime is greater than $k(lnk+lnlnk-1)$ for $k\ge 2$. Math. Comp. 68 (1999) 411-415. | Zbl

[7] H. Edelsbrunner, Algorithms in Combinatorial Geometry. Springer, Berlin (1987). | MR | Zbl

[8] P. Erdös and M. Lewin, $d$-Complete Sequences of Integers. Math. Comp. 65 (1996) 837-840. | Zbl

[9] P.C. Fishburn and F.S. Roberts, Uniqueness in finite measurement, in Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, edited by F.S. Roberts. Springer, New York (1989) 103-137. | Zbl

[10] R.K. Guy, Unsolved Problems in Number Theory. Springer, New York, 2nd ed. (1994). | MR | Zbl

[11] D.E. Knuth, The Art of Computer Programming, Vols. 2 and 3. Addison-Wesley, Reading (1997 and 1998). | MR

[12] J.C. Lagarias, The $3x+1$ problem and its generalizations. American Math. Monthly 92 (1985) 3-23. | Zbl

[13] W.J. Leveque, Fundamentals of Number Theory. Addison-Wesley, Reading (1977). | MR | Zbl

[14] G. Melfi, On Two Conjectures about Practical Numbers. J. Number Theory 56 (1996) 205-210. | Zbl

[15] M.B. Nathanson, Additive Number Theory. Springer, New York (1996). | MR | Zbl

[16] E. Neuwirth, Computing Tournament Sequence Numbers efficiently with Matrix Techniques. Sém. Lothar. Combin. 47 (2002), Article B47h, p. 12 (electronic). | MR | Zbl

[17] A.M. Odlyzko, Iterated Absolute Values of Differences of Consecutive Primes. Math. Comp. 61 (1993) 373-380. | Zbl

[18] D.P. Robbins, The Story of 1, 2, 7, 42, 429, 7436, ... Math. Intellig. 13 (1991) 12-19. | Zbl

[19] J. Richstein, Verifying the Goldbach Conjecture up to $4\times {10}^{14}$. Math. Comp. 70 (2001) 1745-1749. | MR | Zbl

[20] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. World-Wide Web URL http://www.research.att.com/~njas/sequences/

[21] N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences. Academic Press, New York (1995). | MR | Zbl