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A003668 a(n) is smallest number which is uniquely a(j)+a(k), j

%I A003668 M1731 #43 Feb 16 2025 08:32:27
%S A003668 2,7,9,11,13,15,16,17,19,21,25,29,33,37,39,45,47,53,61,69,71,73,75,85,
%T A003668 89,101,103,117,133,135,137,139,141,143,145,147,151,155,159,163,165,
%U A003668 171,173,179,187,195,197,199,201,211,215,227,229,243,259,261,263,265,267,269
%N A003668 a(n) is smallest number which is uniquely a(j)+a(k), j<k.
%C A003668 An Ulam-type sequence - see A002858 for many further references, comments, etc. - _T. D. Noe_, Jan 21 2008
%D A003668 R. K. Guy, "s-Additive sequences", preprint, 1994.
%D A003668 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003668 T. D. Noe, <a href="/A003668/b003668.txt">Table of n, a(n) for n = 1..1000</a>
%H A003668 M. Akeran, <a href="/A003668/a003668.pdf">On some 1-additive sequences</a>
%H A003668 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 A003668 S. R. Finch, <a href="http://www.emis.de/journals/EM/">Patterns in 1-additive sequences</a>, Experimental Mathematics 1 (1992), 57-63.
%H A003668 R. K. Guy, <a href="/A007300/a007300.pdf">s-Additive sequences</a>, Preprint, 1994. (Annotated scanned copy)
%H A003668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UlamSequence.html">Ulam Sequence</a>
%H A003668 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_number">Ulam number</a>
%H A003668 <a href="/index/U#Ulam_num">Index entries for Ulam numbers</a>
%F A003668 Akeran gives a formula.
%F A003668 For n>7, a(n+26)=a(n)+126. - _T. D. Noe_, Jan 21 2008
%t A003668 Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Total, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {2, 7}, 58] (* _Michael De Vlieger_, Nov 16 2017 *)
%o A003668 (Haskell)
%o A003668 a003668 n = a003668_list !! (n-1)
%o A003668 a003668_list = 2 : 7 : ulam 2 7 a003668_list
%o A003668 -- Function ulam as defined in A002858.
%o A003668 -- _Reinhard Zumkeller_, Nov 03 2011
%o A003668 (Python)
%o A003668 def aupton(terms):
%o A003668   alst = [2, 7]
%o A003668   for n in range(2, terms):
%o A003668     sums = [alst[j]+alst[k] for j in range(n-1) for k in range(j+1, n)]
%o A003668     alst.append(min([s for s in sums if sums.count(s)==1 and s > alst[-1]]))
%o A003668   return alst
%o A003668 print(aupton(60)) # _Michael S. Branicky_, Feb 07 2021
%Y A003668 Cf. A100729.
%K A003668 nonn
%O A003668 1,1
%A A003668 _N. J. A. Sloane_, _Mira Bernstein_