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A356088 a(n) = A001951(A022838(n)).

%I A356088 #21 Apr 13 2025 01:46:00
%S A356088 1,4,7,8,11,14,16,18,21,24,26,28,31,33,35,38,41,43,45,48,50,53,55,57,
%T A356088 60,63,65,67,70,72,74,77,80,82,84,87,90,91,94,97,100,101,104,107,108,
%U A356088 111,114,117,118,121,124,127,128,131,134,135,138,141,144,145
%N A356088 a(n) = A001951(A022838(n)).
%C A356088 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 A356088 (1) u o v, defined by (u o v)(n) = u(v(n));
%C A356088 (2) u o v';
%C A356088 (3) u' o v;
%C A356088 (4) u' o v'.
%C A356088 Every positive integer is in exactly one of the four sequences.
%C A356088 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 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
%C A356088 For A356088, 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 A356088 (1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
%e A356088 (2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
%e A356088 (3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
%e A356088 (4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.
%t A356088 z = 600; zz = 100;
%t A356088 u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
%t A356088 u1 = Complement[Range[Max[u]], u];  (* A001952 *)
%t A356088 v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
%t A356088 v1 = Complement[Range[Max[v]], v];  (* A054406 *)
%t A356088 Table[u[[v[[n]]]], {n, 1, zz}];    (* A356088 *)
%t A356088 Table[u[[v1[[n]]]], {n, 1, zz}];   (* A356089 *)
%t A356088 Table[u1[[v[[n]]]], {n, 1, zz}];   (* A356090 *)
%t A356088 Table[u1[[v1[[n]]]], {n, 1, zz}];  (* A356091 *)
%o A356088 (Python)
%o A356088 from math import isqrt
%o A356088 def A356088(n): return isqrt(isqrt(3*n*n)**2<<1) # _Chai Wah Wu_, Aug 06 2022
%Y A356088 Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356089, A356090, A356091.
%K A356088 nonn,easy
%O A356088 1,2
%A A356088 _Clark Kimberling_, Aug 04 2022