A002859 a(1) = 1, a(2) = 3; for n >= 3, a(n) is smallest number that is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

1, 3, 4, 5, 6, 8, 10, 12, 17, 21, 23, 28, 32, 34, 39, 43, 48, 52, 54, 59, 63, 68, 72, 74, 79, 83, 98, 99, 101, 110, 114, 121, 125, 132, 136, 139, 143, 145, 152, 161, 165, 172, 176, 187, 192, 196, 201, 205, 212, 216, 223, 227, 232, 234, 236, 243, 247, 252, 256, 258

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

Comments

An Ulam-type sequence - see A002858 for many further references, comments, etc.

Examples

			7 is missing since 7 = 1 + 6 = 3 + 4; but 8 is present since 8 = 3 + 5 has a unique representation.
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
• R. K. Guy, Unsolved Problems in Number Theory, Section C4.
• R. K. Guy, "s-Additive sequences," preprint, 1994.
• C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 358.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. M. Ulam, Problems in Modern Mathematics, Wiley, NY, 1960, p. ix.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Ulam s-Additive Sequences [From Steven Finch, Apr 20 2019]
• Raymond Queneau, Sur les suites s-additives, J. Combin. Theory A 12(1) (1972), 31-71.
• N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
• S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962. [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, Ulam Sequence.
• Wikipedia, Ulam number.
• Index entries for Ulam numbers.

Crossrefs

Cf. A002858 (version beginning 1,2), A199118, A199119.

Programs

Haskell

a002859 n = a002859_list !! (n-1)
a002859_list = 1 : 3 : ulam 2 3 a002859_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Mathematica

s = {1, 3}; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n-s], n/2, 1, 1]] != 2]; n], {60}]; s (* Jean-François Alcover, Oct 20 2011 *)