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%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
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