A356219 - OEIS

%I #4 Nov 19 2022 20:35:55

%S 284,287,289,292,294,296,299,301,304,306,309,311,313,316,318,321,323,

%T 325,328,330,333,335,337,340,342,345,347,350,352,354,357,359,362,364,

%U 366,369,371,374,376,379,381,383,386,388,391,393,395,398,400

%N Intersection of A001952 and A003151.

%C This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:

%C (1)  u ^ v = intersection of u and v (in increasing order);

%C (2)  u ^ v';

%C (3)  u' ^ v;

%C (4)  u' ^ 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 A356219, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).

%e (1)  u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) =  A003151

%e (2)  u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) =  A001954

%e (3)  u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356219

%e (4)  u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152

%t z = 200;

%t r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)

%t u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)

%t r1 = 1 + Sqrt[2]; v = Table[Floor[n*r1], {n, 1, z}]  (* A003151 *)

%t v1 = Take[Complement[Range[1000], v], z]  (* A003152 *)

%t t1 = Intersection[u, v]    (* A003151 *)

%t t2 = Intersection[u, v1]   (* A001954 *)

%t t3 = Intersection[u1, v]   (* A356219 *)

%t t4 = Intersection[u1, v1]  (* A001952 *)

%Y Cf. A001951, A001952, A003151, A003152, A001954, A184922 (results of compositions instead of intersections), A341239 (reversed compositions).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Nov 13 2022