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A351415
Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).
11
4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, 46, 51, 53, 55, 58, 64, 67, 69, 71, 76, 80, 82, 84, 87, 93, 98, 100, 105, 111, 114, 116, 118, 122, 127, 129, 131, 134, 140, 145, 147, 152, 156, 158, 160, 163, 165, 169, 174, 176, 181, 187, 190, 192, 194, 199
OFFSET
1,1
COMMENTS
Conjecture: every term of the difference sequence is in {2,3,4,5,6}, and each occurs infinitely many times.
From Clark Kimberling, Jul 29 2022: (Start)
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. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1) u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) = A351415
(2) u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, ...) = A356101
(3) u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4) u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, ...) = A356103
(End)
EXAMPLE
The two Beatty sequences are (1,3,4,6,8,9,11,12,14,...) and (2,4,6,8,11,13,15,17,...), with common terms forming the sequence (4,6,8,11,...).
MATHEMATICA
z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}] (* A000201 *)
u1 = Take[Complement[Range[1000], u], z] (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}] (* A022839 *)
v1 = Take[Complement[Range[1000], v], z] (* A108598 *)
Intersection[u, v] (* A351415 *)
Intersection[u, v1] (* A356101 *)
Intersection[u1, v] (* A356102 *)
Intersection[u1, v1] (* A356103 *)
CROSSREFS
Cf. A001950, A108598, A356101, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).
Sequence in context: A343946 A293806 A310661 * A139404 A334927 A334920
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 10 2022
STATUS
approved