login
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
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.
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 26 2022
STATUS
approved