A356183 - OEIS

%I #4 Sep 04 2022 12:55:33

%S 7,14,23,30,40,47,54,63,70,80,87,94,104,111,120,127,137,144,151,160,

%T 167,177,184,191,201,208,217,224,234,241,248,257,264,274,281,288,298,

%U 305,314,321,328,338,345,354,362,371,378,385,395,402,411,418,425,435

%N a(n) = A054406(A001952(n)).

%C This is the fourth 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) v o u, defined by (v o u)(n)  = v(u(n));

%C (2) v' o u;

%C (3) v o u';

%C (4) v' o u'.

%C Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A356088 to A356091.

%C Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo) w(n)/n exists and defines the (limiting) density of w.  For w = u,v,u',v', denote the densities by r,s,r',s'.  Then the densities of sequences (1)-(4) exist, and

%C 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.

%C For A356183, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

%e (1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180

%e (2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181

%e (3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182

%e (4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183

%t z = 800; zz = 100;

%t u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)

%t u1 = Complement[Range[Max[u]], u];       (* A001952 *)

%t v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)

%t v1 = Complement[Range[Max[v]], v];  (* A054406 *)

%t Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)

%t Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181) *)

%t Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)

%t Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)

%Y Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356181, A356182.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Aug 24 2022