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A356217 a(n) = A022839(A000201(n)).

%I A356217 #22 Mar 23 2025 18:24:08
%S A356217 2,6,8,13,17,20,24,26,31,35,38,42,46,49,53,55,60,64,67,71,73,78,82,84,
%T A356217 89,93,96,100,102,107,111,114,118,122,125,129,131,136,140,143,147,149,
%U A356217 154,158,160,165,169,172,176,178,183,187,190,194,196,201,205,207
%N A356217 a(n) = A022839(A000201(n)).
%C A356217 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 A356217 (1) v o u, defined by (v o u)(n) = v(u(n));
%C A356217 (2) v' o u;
%C A356217 (3) v o u';
%C A356217 (4) v' o u'.
%C A356217 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 A356217 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 A356217 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
%C A356217 For A356217 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 A356217 (1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
%e A356217 (2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
%e A356217 (3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
%e A356217 (4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220
%t A356217 z = 1000;
%t A356217 u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
%t A356217 u1 = Complement[Range[Max[u]], u];  (* A001950 *)
%t A356217 v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
%t A356217 v1 = Complement[Range[Max[v]], v];  (* A108598 *)
%t A356217 Table[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
%t A356217 Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
%t A356217 Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
%t A356217 Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)
%o A356217 (Python)
%o A356217 from math import isqrt
%o A356217 def A356217(n): return isqrt(5*(n+isqrt(5*n**2)>>1)**2) # _Chai Wah Wu_, Oct 14 2022
%Y A356217 Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356218, A190509, A356220.
%K A356217 nonn,easy
%O A356217 1,1
%A A356217 _Clark Kimberling_, Oct 02 2022