proposed

approved

editing

proposed

allocated for Clark Kimberling

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

7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, 111, 120, 127, 137, 144, 151, 160, 167, 177, 184, 191, 201, 208, 217, 224, 234, 241, 248, 257, 264, 274, 281, 288, 298, 305, 314, 321, 328, 338, 345, 354, 362, 371, 378, 385, 395, 402, 411, 418, 425, 435

1,1

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:

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

(2) v' o u;

(3) v o u';

(4) v' o u'.

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.

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

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

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.

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

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

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

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

z = 800; zz = 100;

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

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

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

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

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

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

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

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

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

allocated

nonn,easy

Clark Kimberling, Aug 24 2022

approved

editing

allocated for Clark Kimberling

allocated

approved