cp's OEIS Frontend

A356180 a(n) = A022838(A001951(n)).

%I A356180 #12 Jun 05 2023 08:56:02
%S A356180 1,3,6,8,12,13,15,19,20,24,25,27,31,32,36,38,41,43,45,48,50,53,55,57,
%T A356180 60,62,65,67,71,72,74,77,79,83,84,86,90,91,95,96,98,102,103,107,109,
%U A356180 112,114,116,119,121,124,126,128,131,133,136,138,142,143,145,148
%N A356180 a(n) = A022838(A001951(n)).
%C A356180 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 A356180 (1) v o u, defined by (v o u)(n)  = v(u(n));
%C A356180 (2) v' o u;
%C A356180 (3) v o u';
%C A356180 (4) v' o u'.
%C A356180 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 A356088 to A356091.
%C A356180 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 A356180 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
%C A356180 For A356180, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.
%e A356180 (1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
%e A356180 (2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
%e A356180 (3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
%e A356180 (4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183
%t A356180 z = 800; zz = 100;
%t A356180 u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
%t A356180 u1 = Complement[Range[Max[u]], u];       (* A001952 *)
%t A356180 v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
%t A356180 v1 = Complement[Range[Max[v]], v];  (* A054406 *)
%t A356180 Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
%t A356180 Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
%t A356180 Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
%t A356180 Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)
%o A356180 (Python)
%o A356180 from math import isqrt
%o A356180 def A356180(n): return isqrt(3*isqrt(n**2<<1)**2) # _Chai Wah Wu_, Sep 06 2022
%Y A356180 Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356181, A356182, A356183.
%K A356180 nonn,easy
%O A356180 1,2
%A A356180 _Clark Kimberling_, Aug 24 2022