A360395 - OEIS

%I #8 Feb 22 2023 08:07:43

%S 1,6,9,15,19,24,27,31,36,42,45,51,55,60,66,69,73,78,81,87,91,96,99,

%T 103,108,114,117,121,126,129,135,139,144,150,153,159,163,168,171,175,

%U 180,186,189,195,199,204,210,213,217,222,225,231,235,240,246,249,255

%N Intersection of A026430 and A360394.

%C This is the second 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 (1)  u ^ v = intersection of u and v (in increasing order);

%C (2)  u ^ v';

%C (3)  u' ^ v;

%C (4)  u' ^ v'.

%C 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 For A360395, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:

%C u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430

%C u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u

%C v = u + 1 = A285954, except its initial 1

%C v' = complement of v.

%e (1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394

%e (2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395

%e (3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396

%e (4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

%t z = 400;

%t u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)

%t u1 = Complement[Range[Max[u]], u];  (* A356133 *)

%t v = u + 2 ; (* A360392 *)

%t v1 = Complement[Range[Max[v]], v];  (* A360393 *)

%t Intersection[u, v]    (* A360394 *)

%t Intersection[u, v1]   (* A360395 *)

%t Intersection[u1, v]   (* A360396 *)

%t Intersection[u1, v1]  (* A360397 *)

%Y Cf. A026430, A356133, A360392-A360394, A360396-A360405.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 05 2023