A347793 Intersection of Beatty sequences for 2^(1/3) and 2^(2/3).

0, 1, 3, 6, 7, 11, 12, 15, 17, 20, 22, 23, 25, 26, 28, 30, 31, 34, 36, 39, 41, 42, 44, 46, 47, 49, 50, 52, 55, 57, 60, 61, 65, 66, 68, 69, 71, 73, 74, 76, 79, 80, 84, 85, 88, 90, 93, 95, 98, 100, 103, 104, 107, 109, 112, 114, 115, 117, 119, 120, 122, 123

Offset: 0

Author

Clark Kimberling, Nov 01 2021

Keywords

• nonn

Comments

Let d(n) = a(n) - 2n. Conjecture: (d(n)) is unbounded below and above, and d(n) = 0 for infinitely many n.

In general, if r and s are irrational numbers greater than 1, and a(n) is the n-th term of the intersection of the Beatty sequences for r and s, then a(n) = floor(r*ceiling(a(n)/r)) = floor(s*ceiling(a(n)/s)).

Examples

			Beatty sequence for 2^(1/3): (0,1,2,3,5,6,7,8,10,11,...)
Beatty sequence for 2^(2/3): (0,1,3,4,6,7,9,11,12,,...)
Intersection = (0,1,3,6,7,11,12,...).
		

Crossrefs

Cf. A038129, A347792.

Programs

Mathematica

z = 200; r = 2^(1/3); s = 2^(2/3);
u = Table[Floor[n r], {n, 0, z}];  (* A038129 *)
v = Table[Floor[n s], {n, 0, z}];  (* A347792 *)
Intersection[u, v]  (* A347793 *)