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%I #6 Nov 20 2021 21:25:56
%S 0,1,3,6,7,11,12,15,17,20,22,23,25,26,28,30,31,34,36,39,41,42,44,46,
%T 47,49,50,52,55,57,60,61,65,66,68,69,71,73,74,76,79,80,84,85,88,90,93,
%U 95,98,100,103,104,107,109,112,114,115,117,119,120,122,123
%N Intersection of Beatty sequences for 2^(1/3) and 2^(2/3).
%C Let d(n) = a(n) - 2n. Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n.
%C In general, if r and s are irrational numbers greater than 1, and a(n) is the n-th term of the intersection of the Beatty sequences for r and s, then a(n) = floor(r*ceiling(a(n)/r)) = floor(s*ceiling(a(n)/s)).
%e Beatty sequence for 2^(1/3): (0,1,2,3,5,6,7,8,10,11,...)
%e Beatty sequence for 2^(2/3): (0,1,3,4,6,7,9,11,12,,...)
%e Intersection = (0,1,3,6,7,11,12,...).
%t z = 200; r = 2^(1/3); s = 2^(2/3);
%t u = Table[Floor[n r], {n, 0, z}]; (* A038129 *)
%t v = Table[Floor[n s], {n, 0, z}]; (* A347792 *)
%t Intersection[u, v] (* A347793 *)
%Y Cf. A038129, A347792.
%K nonn
%O 0,3
%A _Clark Kimberling_, Nov 01 2021
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