%I #29 Dec 06 2018 16:37:19

%S 2,4,8,15,17,19,21,26,28,32,35,39,41,46,48,52,59,61,63,65,70,72,76,78,

%T 83,85,89,92,96,100,102,107,109,113,116,120,122,127,129,133,140,144,

%U 146,151,153,157,159,164,166,170,173,177,181,184,188,190,195,197,201,203,208,210,212,214,221

%N Numbers not in any of A158919, A277722, A277723.

%C Let tau be the tribonacci constant (A058265). Although 1/tau + 1/tau^2 + 1/tau^3 = 1, by Uspensky's 1927 theorem, the three sequences floor(n*tau) (A158919), floor(n*tau^2) (A277722), and floor(n*tau^3) (A277723) cannot form a partition of the nonnegative integers. (Compare Beatty's theorem.)

%C Entries A277724-A277727 investigate how these three sequences meet, and the present sequence gives the numbers not in any of the three sequences. Any two of the three sequences have a nontrivial intersection, while the intersection of all three is {0}.

%C On the other hand, the three sequences A003144, A003145, A003146, which arise from the tribonacci word, DO form a partition of the positive integers and are closely connected with the three sequences mentioned in the definition.

%C It would be nice to have b-files for all the sequences mentioned here. (Many do, but some do not.)

%H Jean-François Alcover, <a href="/A277728/b277728.txt">Table of n, a(n) for n = 1..2955</a>

%H S. Beatty, A. Ostrowski, J. Hyslop, and A. C. Aitken, <a href="http://www.jstor.org/stable/2298716">Problems and Solutions: Solutions: 3177</a>, Amer. Math. Monthly, 34 (1927), pp. 159-160.

%H R. L. Graham, <a href="http://www.jstor.org/stable/2311859">On a theorem of Uspensky</a>, Amer. Math. Mnthly, 70 (1963): 407-409.

%H A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.

%H J. V. Uspensky, <a href="http://www.jstor.org/stable/2299838">On a problem arising out of the theory of a certain game</a>, Amer. Math. Mnthly., 34 (1927), 516-521.

%t maxTerm = 10000;

%t a19[n_] := Floor[n*Root[#^3 - #^2 - # - 1&, 1]];

%t a22[n_] := Floor[n*Root[#^3 - 3 #^2 - # - 1&, 1]];

%t a23[n_] := Floor[n*Root[#^3 - 7 #^2 + 5 # - 1&, 1]];

%t A19 = Reap[k = 1; While[a19[k] <= maxTerm, Sow[a19[k++]]]][[2, 1]];

%t A22 = Reap[k = 1; While[a22[k] <= maxTerm, Sow[a22[k++]]]][[2, 1]];

%t A23 = Reap[k = 1; While[a23[k] <= maxTerm, Sow[a23[k++]]]][[2, 1]];

%t Select[Range[maxTerm], FreeQ[A19, #] && FreeQ[A22, #] && FreeQ[A23, #]&] (* _Jean-François Alcover_, Dec 06 2018 *)

%Y Cf. A003144, A003145, A003146, A058265, A158919, A184820, A277722, A277723, A277724, A277725, A277726, A277727.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Oct 30 2016