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A356052 Intersection of A001951 and A137803.

%I A356052 #10 Mar 23 2025 03:20:58
%S A356052 1,5,7,9,11,15,19,21,22,24,26,28,32,36,38,42,45,49,53,55,57,59,63,65,
%T A356052 66,70,72,74,76,80,82,84,86,89,91,93,97,101,103,107,111,114,118,120,
%U A356052 124,128,130,132,135,137,141,145,147,149,151,155,156,158,162,164
%N A356052 Intersection of A001951 and A137803.
%C A356052 This is the first 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 A356052 (1)  u ^ v = intersection of u and v (in increasing order);
%C A356052 (2)  u ^ v';
%C A356052 (3)  u' ^ v;
%C A356052 (4)  u' ^ v'.
%C A356052 Every positive integer is in exactly one of the four sequences.
%C A356052 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 A356052 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
%C A356052 For A356052, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.
%e A356052 (1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) = A356052
%e A356052 (2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) = A356053
%e A356052 (3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
%e A356052 (4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055
%t A356052 z = 250;
%t A356052 u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
%t A356052 u1 = Complement[Range[Max[u]], u]     (* A001952 *)
%t A356052 v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
%t A356052 v1 = Complement[Range[Max[v]], v]     (* A137804 *)
%t A356052 Intersection[u, v]    (* A356052 *)
%t A356052 Intersection[u, v1]   (* A356053 *)
%t A356052 Intersection[u1, v]   (* A356054 *)
%t A356052 Intersection[u1, v1]  (* A356055 *)
%Y A356052 Cf. A001951, A001952, A136803, A137804, A356053, A356054, A356055, A356056 (composites instead of intersections), A356081.
%K A356052 nonn,easy
%O A356052 1,2
%A A356052 _Clark Kimberling_, Jul 26 2022