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A360394 Intersection of A026430 and A360392.

%I A360394 #8 Feb 22 2023 08:07:40
%S A360394 3,5,8,10,12,14,16,18,21,23,26,28,30,33,35,37,39,41,44,46,48,50,52,54,
%T A360394 57,59,61,63,65,68,70,72,75,77,80,82,84,86,88,90,93,95,98,100,102,105,
%U A360394 107,109,111,113,116,118,120,123,125,128,130,132,134,136,138
%N A360394 Intersection of A026430 and A360392.
%C A360394 This is the first 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 A360394 (1)  u ^ v = intersection of u and v (in increasing order);
%C A360394 (2)  u ^ v';
%C A360394 (3)  u' ^ v;
%C A360394 (4)  u' ^ v'.
%C A360394 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 (and likewise for A360402-A360405).
%C A360394 For A360394, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
%C A360394 u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
%C A360394 u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
%C A360394 v = u + 1 = A285954, except its initial 1
%C A360394 v' = complement of v.
%e A360394 (1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
%e A360394 (2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
%e A360394 (3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
%e A360394 (4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397
%t A360394 z = 400;
%t A360394 u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
%t A360394 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360394 v = u + 2 ; (* A360392 *)
%t A360394 v1 = Complement[Range[Max[v]], v];  (* A360393 *)
%t A360394 Intersection[u, v]     (* A360394 *)
%t A360394 Intersection[u, v1]    (* A360395 *)
%t A360394 Intersection[u1, v]    (* A360396 *)
%t A360394 Intersection[u1, v1]   (* A360397 *)
%Y A360394 Cf. A026430, A356133, A360392, A360393, A356850-A360405.
%K A360394 nonn,easy
%O A360394 1,1
%A A360394 _Clark Kimberling_, Feb 05 2023