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Size of the largest set encompassing no {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, ...} (A003586).
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%I #28 Mar 06 2020 00:01:49

%S 1,2,2,3,4,5,5,6,7,7,8,8,9,10,11,11,12,13,13,14,14,15,16,17,17,18,18,

%T 19,20,21,21,22,22,23,24,25,25,26,26,27,28,29,30,30,31,31,32,33,33,34,

%U 35,35,36,36,37,38,39,39,40,41,42,42,42,43,44,45,45,46,47

%N Size of the largest set encompassing no {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, ...} (A003586).

%C This is the weakly triple-free analog of A157271.

%C A094708(n) = n - a(n).

%C Position of first n gives A004059(n).

%C Graham paper erroneously has a(30)=20. - _Sean A. Irvine_, Nov 18 2015

%D R. L. Graham et al., On extremal density theorems for linear forms, pp. 103-109 of H. Zassenhaus, editor, Number Theory and Algebra. Academic Press, NY, 1977.

%H F. Chung, P. Erdős, R. Graham, <a href="http://www.math.ucsd.edu/~ronspubs/02_04_linear_forms.pdf">On sparse sets hitting linear forms</a>, Proc. Number Theory for the Millennium, 1, pp. 257-272, 2000.

%H Steven R. Finch, <a href="/FinchTriple.html">Triple-Free Sets of Integers</a> [From Steven Finch, Apr 20 2019]

%H R. L. Graham et al., <a href="http://www.math.ucsd.edu/~ronspubs/77_05_extremal_density.pdf">On extremal density theorems for linear forms</a>, pp. 103-109 of H. Zassenhaus, editor, Number Theory and Algebra. Academic Press, NY, 1977.

%e A set for a(30) is {1, 2, 6, 8, 9, 12, 16, 27, 36, 48, 54, 64, 72, 96, 128, 162, 216, 243, 256, 288, 324}. - _Sean A. Irvine_, Oct 26 2015

%Y Cf. A004059, A094708, A003586, A157271.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _Steven Finch_, Feb 25 2009

%E Revised by _N. J. A. Sloane_, Jun 13 2012

%E a(30) corrected and more terms from _Sean A. Irvine_, Oct 26 2015