%I #14 Nov 18 2022 07:15:42
%S 1,5,7,10,14,16,19,21,25,28,30,34,37,39,43,45,48,52,54,57,59,63,66,68,
%T 72,75,77,81,83,86,90,92,95,99,101,104,106,110,113,115,119,121,124,
%U 128,130,133,137,139,142,144,148,151,153,157,159,162,166,168,171
%N a(n) = A108598(A000201(n)).
%C This is the second 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) u o v';
%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 A356104 to A356107.
%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 A356218, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
%e (1) v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
%e (2) v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
%e (3) v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
%e (4) v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220
%t z = 1000;
%t u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}]; (* A000201 *)
%t u1 = Complement[Range[Max[u]], u]; (* A001950 *)
%t v = Table[Floor[n*Sqrt[5]], {n, 1, z}]; (* A022839 *)
%t v1 = Complement[Range[Max[v]], v]; (* A108598 *)
%t zz = 120;
%t Table[v[[u[[n]]]], {n, 1, z/4}] (* A356217 *)
%t Table[v1[[u[[n]]]], {n, 1, z/4}] (* A356218 *)
%t Table[v[[u1[[n]]]], {n, 1, z/4}] (* A190509 *)
%t Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)
%Y Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356217, A190509, A356220.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Oct 02 2022