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. (Formerly M2303 N0909)

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

1,2

Comments

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A051916 A130216 A120162 * A180646 A300217 A062514

Adjacent sequences: A002856 A002857 A002858 * A002860 A002861 A002862

Keyword

nonn,nice

Author

N. J. A. Sloane, Mira Bernstein

Status

approved