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%I #11 Aug 06 2022 07:25:02
%S 6,13,23,30,37,47,54,61,71,78,88,95,102,112,119,126,136,143,150,160,
%T 167,177,184,191,201,208,215,225,232,238,249,256,266,273,279,290,297,
%U 303,314,320,331,338,344,355,361,368,378,385,392,402,409,419,426,433
%N a(n) = A001952(A054406(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) 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 A356091, 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.
%e (1) u o v = (1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088
%e (2) u o v' = (2, 5, 9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089
%e (3) u' o v = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090
%e (4) u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091
%t z = 600; zz = 100;
%t u = Table[Floor[n*Sqrt[2]], {n, 1, z}]; (* A001951 *)
%t u1 = Complement[Range[Max[u]], u]; (* A001952 *)
%t v = Table[Floor[n*Sqrt[3]], {n, 1, z}]; (* A022838 *)
%t v1 = Complement[Range[Max[v]], v]; (* A054406 *)
%t Table[u[[v[[n]]]], {n, 1, zz}] (* A356088 *)
%t Table[u[[v1[[n]]]], {n, 1, zz}] (* A356089 *)
%t Table[u1[[v[[n]]]], {n, 1, zz}] (* A356090 *)
%t Table[u1[[v1[[n]]]], {n, 1, zz}] (* A356091 *)
%Y Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356089, A356090.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Aug 05 2022
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