%I #8 Aug 03 2022 23:27:27

%S 1,4,7,9,12,15,18,21,24,26,29,31,33,36,39,42,45,48,50,53,56,59,62,63,

%T 66,69,72,74,77,80,83,86,89,91,93,96,98,101,104,107,110,113,115,118,

%U 121,124,125,128,131,134,137,140,142,145,148,151,154,156,158,161

%N a(n) = A001951(A137803(n)).

%C This is the first 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) u' o v'.

%C Every positive integer is in exactly one of the four sequences.

%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 A356056, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

%F a(n) = A001951(A137803(n)).

%e (1) u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056

%e (2) u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057

%e (3) u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058

%e (4) u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

%t z = 800;

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

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

%t v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}] (* A137803 *)

%t v1 = Complement[Range[Max[v]], v] (* A137804 *)

%t Table[u[[v[[n]]]], {n, 1, z/8}]; (* A356056 *)

%t Table[u[[v1[[n]]]], {n, 1, z/8}]; (* A356057 *)

%t Table[u1[[v[[n]]]], {n, 1, z/8}]; (* A356058 *)

%t Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

%Y Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356057, A356058, A356059.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jul 26 2022