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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.

%I A002859 M2303 N0909 #45 Feb 16 2025 08:32:27
%S A002859 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,
%T A002859 83,98,99,101,110,114,121,125,132,136,139,143,145,152,161,165,172,176,
%U A002859 187,192,196,201,205,212,216,223,227,232,234,236,243,247,252,256,258
%N 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.
%C A002859 An Ulam-type sequence - see A002858 for many further references, comments, etc.
%D A002859 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
%D A002859 R. K. Guy, Unsolved Problems in Number Theory, Section C4.
%D A002859 R. K. Guy, "s-Additive sequences," preprint, 1994.
%D A002859 C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 358.
%D A002859 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002859 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002859 S. M. Ulam, Problems in Modern Mathematics, Wiley, NY, 1960, p. ix.
%H A002859 T. D. Noe, <a href="/A002859/b002859.txt">Table of n, a(n) for n = 1..10000</a>
%H A002859 Steven R. Finch, <a href="/FinchSadd.html">Ulam s-Additive Sequences</a> [From Steven Finch, Apr 20 2019]
%H A002859 Raymond Queneau, <a href="https://doi.org/10.1016/0097-3165(72)90083-0">Sur les suites s-additives</a>, J. Combin. Theory A 12(1) (1972), 31-71.
%H A002859 N. J. A. Sloane, <a href="/A001149/a001149.pdf">Handwritten notes on Self-Generating Sequences, 1970</a> (note that A1148 has now become A005282)
%H A002859 S. M. Ulam, <a href="/A002858/a002858.pdf">On some mathematical problems connected with patterns of growth of figures</a>, 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]
%H A002859 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UlamSequence.html">Ulam Sequence</a>.
%H A002859 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_number">Ulam number</a>.
%H A002859 <a href="/index/U#Ulam_num">Index entries for Ulam numbers</a>.
%e A002859 7 is missing since 7 = 1 + 6 = 3 + 4; but 8 is present since 8 = 3 + 5 has a unique representation.
%t A002859 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 *)
%o A002859 (Haskell)
%o A002859 a002859 n = a002859_list !! (n-1)
%o A002859 a002859_list = 1 : 3 : ulam 2 3 a002859_list
%o A002859 -- Function ulam as defined in A002858.
%o A002859 -- _Reinhard Zumkeller_, Nov 03 2011
%Y A002859 Cf. A002858 (version beginning 1,2), A199118, A199119.
%K A002859 nonn,nice
%O A002859 1,2
%A A002859 _N. J. A. Sloane_, _Mira Bernstein_