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 A356052 Intersection of A001951 and A137803. 10
 1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, 32, 36, 38, 42, 45, 49, 53, 55, 57, 59, 63, 65, 66, 70, 72, 74, 76, 80, 82, 84, 86, 89, 91, 93, 97, 101, 103, 107, 111, 114, 118, 120, 124, 128, 130, 132, 135, 137, 141, 145, 147, 149, 151, 155, 156, 158, 162, 164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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: (1) u ^ v = intersection of u and v (in increasing order); (2) u ^ v'; (3) u' ^ v; (4) u' ^ v'. Every positive integer is in exactly one of the four sequences. 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 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1. 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. LINKS Table of n, a(n) for n=1..60. EXAMPLE (1) u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) = A356052 (2) u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) = A356053 (3) u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054 (4) u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055 MATHEMATICA z = 250; u = Table[Floor[n (Sqrt[2])], {n, 1, z}] (* A001951 *) u1 = Complement[Range[Max[u]], u] (* A001952 *) v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}] (* A137803 *) v1 = Complement[Range[Max[v]], v] (* A137804 *) Intersection[u, v] (* A356052 *) Intersection[u, v1] (* A356053 *) Intersection[u1, v] (* A356054 *) Intersection[u1, v1] (* A356055 *) CROSSREFS Cf. A001951, A001952, A136803, A137804, A356053, A356054, A356055, A356056 (composites instead of intersections), A356081. Sequence in context: A309747 A080384 A086398 * A023380 A076190 A028885 Adjacent sequences: A356049 A356050 A356051 * A356053 A356054 A356055 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 26 2022 STATUS approved

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Last modified June 5 01:39 EDT 2023. Contains 363130 sequences. (Running on oeis4.)