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%I #11 Jun 05 2023 08:56:08
%S 3,6,9,12,17,21,24,27,32,35,38,42,46,50,53,56,61,64,67,71,74,79,82,85,
%T 88,93,97,100,103,108,111,114,118,122,126,129,132,135,140,144,147,150,
%U 155,158,161,165,169,173,176,179,184,187,190,194,197,202,205,208
%N a(n) = A000201(A022839(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 u';
%C (3) u' o v;
%C (4) u' o v'.
%C Every positive integer is in exactly one of the four sequences. For the reverse composites, v o u, v' o u, v o u', v' o u', see A356217 to A356220.
%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 A356104, 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) u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
%e (2) u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
%e (3) u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
%e (4) u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107
%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[u[[v[[n]]]], {n, 1, zz}] (* A356104 *)
%t Table[u[[v1[[n]]]], {n, 1, zz}] (* A356105 *)
%t Table[u1[[v[[n]]]], {n, 1, zz}] (* A356106 *)
%t Table[u1[[v1[[n]]]], {n, 1, zz}] (* A356107 *)
%Y Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356105, A356106, A356107, A351415 (intersections), A356217 (reverse composites).
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Sep 08 2022
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