A360400 - OEIS

%I #4 Mar 24 2023 17:57:59

%S 7,13,20,22,29,32,34,40,47,49,53,58,62,67,74,76,83,85,89,94,97,104,

%T 110,112,115,122,127,131,137,140,142,148,155,157,161,166,169,176,182,

%U 184,187,193,200,202,208,211,215,220,223,229,236,238,244,247,251,257

%N a(n) = A356133(A360392(n)).

%C This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:

%C (1) u o v, defined by (u o v)(n)  = u(v(n));

%C (2) u o v';

%C (3) u' o v;

%C (4) v' o u'.

%C Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

%e (1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398

%e (2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399

%e (3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400

%e (4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

%t z = 2000;

%t u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)

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

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

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

%t zz = 100;

%t Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)

%t Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)

%t Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)

%t Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

%Y Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Mar 11 2023