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A001857 a(1)=2, 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 A001857 M0634 N0231 #47 Feb 16 2025 08:32:24
%S A001857 2,3,5,7,8,9,13,14,18,19,24,25,29,30,35,36,40,41,46,51,56,63,68,72,73,
%T A001857 78,79,83,84,89,94,115,117,126,153,160,165,169,170,175,176,181,186,
%U A001857 191,212,214,230,235,240,245,266,273,278,283,288,325,331,332,337,342
%N A001857 a(1)=2, 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 A001857 An Ulam-type sequence - see A002858 for many further references, comments, etc.
%C A001857 A plot of the first 10^6 terms shows a nearly straight line having a slope of about 11.1. In contrast to A002858, this sequence has many consecutive numbers; of the first 10^6 terms, consecutive numbers appear 141674 times! - _T. D. Noe_, Jan 21 2008
%D A001857 S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
%D A001857 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
%D A001857 R. K. Guy, Unsolved Problems in Number Theory, Section C4.
%D A001857 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001857 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001857 S. M. Ulam, Problems in Modern Mathematics, Wiley, NY, 1960, p. ix.
%H A001857 T. D. Noe, <a href="/A001857/b001857.txt">Table of n, a(n) for n = 1..10000</a>
%H A001857 Steven R. Finch, <a href="/FinchSadd.html">Ulam s-Additive Sequences</a> [From Steven Finch, Apr 20 2019]
%H A001857 J. Cassaigne and S. R. Finch, <a href="http://www.emis.de/journals/EM/expmath/volumes/4/4.html">A class of 1-additive sequences and additive recurrences</a>
%H A001857 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 A001857 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UlamSequence.html">Ulam Sequence</a>
%H A001857 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_number">Ulam number</a>
%H A001857 <a href="/index/U#Ulam_num">Index entries for Ulam numbers</a>
%t A001857 s = {2, 3}; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n-s], n/2, 1, 1]] != 2]; n], {100}]; s (* _Jean-François Alcover_, Sep 08 2011 *)
%o A001857 (Haskell)
%o A001857 a001857 n = a001857_list !! (n-1)
%o A001857 a001857_list = 2 : 3 : ulam 2 3 a001857_list
%o A001857 -- Function ulam as defined in A002858.
%o A001857 -- _Reinhard Zumkeller_, Nov 03 2011
%Y A001857 Cf. A100729.
%Y A001857 Cf. A199122, A199123.
%K A001857 nonn,nice
%O A001857 1,1
%A A001857 _N. J. A. Sloane_
%E A001857 More terms from _Jud McCranie_