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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).
(Formerly M2544)
3
1, 3, 6, 11, 19, 31, 43, 63, 80, 110, 138, 169, 202, 241, 288, 330 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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

From Jon E. Schoenfield, Aug 24 2009: (Start)

Lexicographically first basis that yields a(n) for n = 2..13:

a(2) = 1 from {0 1}

a(3) = 3 from {0 1 2}

a(4) = 6 from {0 1 2 4}

a(5) = 11 from {0 1 2 4 7}

a(6) = 19 from {0 1 2 4 7 12}

a(7) = 31 from {0 1 2 4 8 13 18}

a(8) = 43 from {0 1 2 4 8 14 19 24}

a(9) = 63 from {0 1 2 4 8 15 24 29 34}

a(10) = 80 from {0 1 2 4 8 15 24 29 34 46}

a(11) = 110 from {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}

a(13) = 169 from {0 1 2 5 16 30 38 47 59 65 71 78 91}

(End)

From Lars Blomberg, Oct 31 2015: (Start)

Lexicographically first basis that yields a(n) for n=14..16:

a(14) = 202 from {0,1,2,4,7,24,38,47,56,66,74,82,95,107}

a(15) = 241 from {0,1,2,22,26,36,43,50,82,90,95,98,101,113,128}

a(16) = 288 from {0,4,5,10,31,43,55,58,92,100,111,120,122,129,136,152}

(End)

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=2..17.

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404 (v_alpha).

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs

Z. Skupien, A. Zak, Pair-sums packing and rainbow cliques, in Topics In Graph Theory, 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).

CROSSREFS

Cf. A004135, A004136. See A232234 for a slight variation.

Sequence in context: A332446 A116100 A295066 * A180415 A050228 A114089

Adjacent sequences:  A004130 A004131 A004132 * A004134 A004135 A004136

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(11) from R. J. Mathar, Mar 17 2007

Two more terms from Jon E. Schoenfield, Aug 24 2009

202 and 241 from Skupien et al. - N. J. A. Sloane, Nov 24 2013

a(16) from Lars Blomberg, Oct 31 2015

a(17) from Fausto A. C. Cariboni, Oct 24 2017

STATUS

approved

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Last modified April 6 12:08 EDT 2020. Contains 333273 sequences. (Running on oeis4.)