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A004133 Additive bases: a(n) is the least integer such that there is an n-element set of nonnegative integers, the sums of pairs (of distinct elements) of which are distinct and at most a(n).

%I A004133 M2544 #51 May 04 2020 10:52:35
%S A004133 1,3,6,11,19,31,43,63,80,110,138,169,202,241,288,330
%N A004133 Additive bases: a(n) is the least integer such that there is an n-element set of nonnegative integers, the sums of pairs (of distinct elements) of which are distinct and at most a(n).
%C A004133 a(11) = 110 from the basis {0 1 2 4 8 15 24 29 34 46 64}. a(12)<=138 from {0 1 2 4 19 30 37 42 50 58 64 74} or {0 1 2 7 12 22 37 40 54 63 67 71} or {0 2 4 18 26 34 49 54 55 61 64 74}, for example. a(13) <= 169 from {0 1 2 5 16 30 38 47 59 65 71 78 91} or {0 1 2 5 18 28 35 50 59 65 71 79 90}. a(14) <= 202 from {0 1 2 4 7 24 38 47 56 66 74 82 95 107}. a(15) <= 250 from {0 1 2 4 13 40 61 67 83 90 98 108 113 118 132}. - _R. J. Mathar_, Mar 17 2007
%C A004133 From _Jon E. Schoenfield_, Aug 24 2009: (Start)
%C A004133 Lexicographically first basis that yields a(n) for n = 2..13:
%C A004133 a(2) = 1 from {0 1}
%C A004133 a(3) = 3 from {0 1 2}
%C A004133 a(4) = 6 from {0 1 2 4}
%C A004133 a(5) = 11 from {0 1 2 4 7}
%C A004133 a(6) = 19 from {0 1 2 4 7 12}
%C A004133 a(7) = 31 from {0 1 2 4 8 13 18}
%C A004133 a(8) = 43 from {0 1 2 4 8 14 19 24}
%C A004133 a(9) = 63 from {0 1 2 4 8 15 24 29 34}
%C A004133 a(10) = 80 from {0 1 2 4 8 15 24 29 34 46}
%C A004133 a(11) = 110 from {0 1 2 4 8 15 24 29 34 46 64}
%C A004133 a(12) = 138 from {0 1 2 4 19 30 37 42 50 58 64 74}
%C A004133 a(13) = 169 from {0 1 2 5 16 30 38 47 59 65 71 78 91}
%C A004133 (End)
%C A004133 From _Lars Blomberg_, Oct 31 2015: (Start)
%C A004133 Lexicographically first basis that yields a(n) for n=14..16:
%C A004133 a(14) = 202 from {0,1,2,4,7,24,38,47,56,66,74,82,95,107}
%C A004133 a(15) = 241 from {0,1,2,22,26,36,43,50,82,90,95,98,101,113,128}
%C A004133 a(16) = 288 from {0,4,5,10,31,43,55,58,92,100,111,120,122,129,136,152}
%C A004133 (End)
%C A004133 Lexicographically first basis that yields a(17) = 330 is {0,5,9,10,11,43,62,75,88,112,115,129,136,143,151,159,171}. - _Fausto A. C. Cariboni_, Oct 24 2017
%D A004133 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004133 R. L. Graham and N. J. A. Sloane, <a href="https://doi.org/10.1137/0601045">On Additive Bases and Harmonious Graphs</a>, SIAM J. Algebraic and Discrete Methods, 1 (1980), pp. 382-404 (v_alpha).
%H A004133 R. L. Graham and N. J. A. Sloane, <a href="http://neilsloane.com/doc/RLG/073.pdf">On Additive Bases and Harmonious Graphs</a> [<a href="http://www.math.ucsd.edu/~ronspubs/80_07_harmonious.pdf">alternate link</a>]
%H A004133 Z. Skupien, A. Zak, Pair-sums packing and rainbow cliques, in <a href="http://www.math.uiuc.edu/~kostochk/Zykov90-Topics_in_Graph_Theory.pdf">Topics In Graph Theory</a>, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).
%Y A004133 Cf. A004135, A004136. See A232234 for a slight variation.
%K A004133 nonn,nice,more
%O A004133 2,2
%A A004133 _N. J. A. Sloane_
%E A004133 a(11) from _R. J. Mathar_, Mar 17 2007
%E A004133 Two more terms from _Jon E. Schoenfield_, Aug 24 2009
%E A004133 202 and 241 from Skupien et al. - _N. J. A. Sloane_, Nov 24 2013
%E A004133 a(16) from _Lars Blomberg_, Oct 31 2015
%E A004133 a(17) from _Fausto A. C. Cariboni_, Oct 24 2017