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a(n) = A108598(A001950(n)).
8

%I #14 Jun 05 2023 08:55:57

%S 3,9,12,18,23,27,32,36,41,47,50,56,61,65,70,74,79,85,88,94,97,103,108,

%T 112,117,123,126,132,135,141,146,150,155,161,164,170,173,179,184,188,

%U 193,197,202,208,211,217,222,226,231,235,240,246,249,255,258,264

%N a(n) = A108598(A001950(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) 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 A356220, 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 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. A000201, A001950, A022839, A108598, A351415 (intersections), A356104 (reverse composites), A356217, A356218, A356219.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Nov 13 2022