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A360396 Intersection of A356133 and A360392.

%I A360396 #8 Feb 22 2023 08:07:48
%S A360396 7,11,17,20,25,29,32,38,43,47,53,56,62,67,71,74,79,83,89,92,97,101,
%T A360396 104,110,115,119,122,127,131,137,140,146,151,155,161,164,169,173,176,
%U A360396 182,187,191,197,200,206,211,215,218,223,227,233,236,242,247,251,257
%N A360396 Intersection of A356133 and A360392.
%C A360396 This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
%C A360396 (1)  u ^ v = intersection of u and v (in increasing order);
%C A360396 (2)  u ^ v';
%C A360396 (3)  u' ^ v;
%C A360396 (4)  u' ^ v'.
%C A360396 Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
%C A360396 For A360396, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
%C A360396 u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
%C A360396 u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
%C A360396 v = u + 1 = A285954, except its initial 1
%C A360396 v' = complement of v.
%e A360396 (1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
%e A360396 (2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
%e A360396 (3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
%e A360396 (4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397
%t A360396 z = 400;
%t A360396 u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
%t A360396 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360396 v = u + 2 ; (* A360392 *)
%t A360396 v1 = Complement[Range[Max[v]], v];  (* A360393 *)
%t A360396 Intersection[u, v]    (* A360394 *)
%t A360396 Intersection[u, v1]   (* A360395 *)
%t A360396 Intersection[u1, v]   (* A360396 *)
%t A360396 Intersection[u1, v1]  (* A360397 *)
%Y A360396 Cf. A026430, A356133, A360392-A360395, A360397-A360405.
%K A360396 nonn,easy
%O A360396 1,1
%A A360396 _Clark Kimberling_, Feb 05 2023