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A360397 Intersection of A356133 and A360393.

%I A360397 #10 Feb 22 2023 12:35:25
%S A360397 2,4,13,22,34,40,49,58,64,76,85,94,106,112,124,133,142,148,157,166,
%T A360397 178,184,193,202,208,220,229,238,244,253,262,274,280,292,301,310,322,
%U A360397 328,337,346,352,364,373,382,394,400,412,421,430,436,445,454,466,472
%N A360397 Intersection of A356133 and A360393.
%C A360397 This is the fourth 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 A360397 (1)  u ^ v = intersection of u and v (in increasing order);
%C A360397 (2)  u ^ v';
%C A360397 (3)  u' ^ v;
%C A360397 (4)  u' ^ v'.
%C A360397 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 A360397 For A360397, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
%C A360397 u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430;
%C A360397 u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u;
%C A360397 v = u + 1 = A285954, except its initial 1;
%C A360397 v' = complement of v.
%e A360397 (1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
%e A360397 (2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
%e A360397 (3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
%e A360397 (4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397
%t A360397 z = 400;
%t A360397 u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
%t A360397 u1 = Complement[Range[Max[u]], u];  (* A356133 *)
%t A360397 v = u + 2 ; (* A360392 *)
%t A360397 v1 = Complement[Range[Max[v]], v];    (* A360393 *)
%t A360397 Intersection[u, v]    (* A360394 *)
%t A360397 Intersection[u, v1]   (* A360395 *)
%t A360397 Intersection[u1, v]   (* A360396 *)
%t A360397 Intersection[u1, v1]  (* A360397 *)
%Y A360397 Cf. A026430, A356133, A360392-A360396, A360398-A360405.
%K A360397 nonn,easy
%O A360397 1,1
%A A360397 _Clark Kimberling_, Feb 10 2023